in this video we're going to cover a few common concepts that you'll see in a typical algebra course so let's get right into it so let's say if you have two terms 5x plus 4x these are considered like terms whenever you have like terms you're allowed to add the coefficients 5 plus 4 is 9 so 5x plus 4x is 9x now let's see if you have another expression 3x plus 4y plus 5x plus 8y you can't add 5x and 8y because they're not like terms however you can add 3x and 5x that's going to equal 8x and you can also add 4y and 8y that's going to be 12y so you can add the coefficients of like terms another example let's say if you have three radical two plus five radical seven plus eight radical two plus three radical seven these are like terms because the radical two is the same you can add the three and five not three and five but the three and eight three plus eight is eleven so it's eleven root two now these two are like terms you can add the five and the three to give you eight so it's eight root seven try this one seven x plus 4x squared plus 5x plus 9x squared so we can add the 4x squared and a 9x squared because they're like terms they both have an x squared four plus nine is thirteen x squared we can add the seven x and a five x seven and five is twelve so this is the answer try this one let's say if you have nine x squared plus six x plus five plus three x squared minus five x minus nine so notice that we can add 9x squared and 3x squared they're like terms so that's going to be 12 x squared now we can also add 6x and 5x 6x plus negative 5x is the same as six x minus five x six minus five is one so we're just going to get one x five plus negative nine five minus nine is negative four so the final answer is 12 x squared plus x minus four here's another example you can try three x squared plus seven x minus four minus eight x squared minus five x plus seven now notice that we have a negative sign we need to distribute the negative sign to the three terms on the right so for the three numbers on the left we don't have anything in front of the parentheses so we could simply open it we don't need the parentheses on the left side so it's simply three x squared plus seven x minus four now you can treat this as a negative one so let's distribute the negative sign to everything on the right so instead of having positive 8x squared is going to be negative 8x squared and instead of having negative 5x it's going to be a positive 5x and instead of having positive seven if we distribute the negative sign to it it's negative seven and so now let's add like terms three x squared plus negative eight x squared three minus eight is negative five so it's negative five x squared seven x plus five x seven plus five is twelve so it's twelve x negative four plus negative seven is the same as negative 4 minus 7 which is negative 11 and so that's how you can add or subtract polynomials so what is a polynomial a polynomial is a function with many terms a monomial is simply one term 8x is a monomial 5x squared that's a monomial or three x squared y is a monomial a binomial has two terms like five x plus six seven x minus three those are binomials a trinomial has three terms so x squared plus six x plus five is a trinomial a polynomial is an expression that has many terms now let's say if we wish to multiply a monomial by a trinomial what is the answer what do we need to do so we need to distribute 7x to everything inside so what's 7x times x squared so x squared is the same as 1x squared 7 times 1 is 7. now what's x times x squared x is the same as x to the first power times x squared whenever you multiply variables you need to add the exponents one plus two is three so this is x cubed x to the first power is simply x x squared is x times x together you have a total of three x's multiplied to each other that's why it's x cubed therefore seven x times x squared is seven x cubed now what is seven x times two x seven times two is fourteen x times x or x to the first power times x to the first power one plus one is two so it's x squared and then seven x times negative three is negative twenty one x so that's how you can multiply a monomial by a trinomial so now it's your turn try this 5x squared times 3x to the fourth power minus 6x cubed plus 5x minus eight so let's distribute the five x squared five x squared times three x to the fourth five times three is fifteen x squared times x to the fourth is x to the sixth we need to add two and four two plus four is six so now we need to multiply five x squared times negative six x cubed five times negative six is negative thirty x squared times x cubed two plus three is five so it's going to be x to the fifth power and then five x squared times five x five times five is twenty five x squared times x we have two plus one which is three and so it's simply going to be x cubed and then five x squared times negative eight that's five times negative eight is negative forty so it's negative forty x squared so this is the answer to this problem now let's multiply a binomial by another binomial so what's 3x minus 4 times two x plus seven so what we need to do here is something called foil we need to foil these two expressions what's three x times two x three times two is six and x times x is x squared keep minus is x to the first power one plus one is two so next we need to multiply the three x by the seven three x times 7 is simply 21x negative 4 times 2x is negative 8x and negative 4 times 7 that's going to be negative 28. so now we need to combine like terms the like terms that we have in this expression is 21x and negative 8x so what's 21 minus 8 21 minus 8 is positive 13 or positive 13x so this is the final answer that's how you can foil two binomial expressions so the answer is 6x squared plus 13x minus 28. try this example 2x minus 5 and 4x plus 7. go ahead and foil this expression so 2x times 4x that's 8x squared and 2x times 7 is 14x negative 5 and 4x is negative 20x and negative 5 times 7 is negative 35. so let's add like terms 14x minus 20x is negative 6x so this is the final answer to this particular problem now what if you see an expression that looks like this what's 2x minus 3 squared how would you simplify this expression so 2x minus 3 squared is the same as 2x minus 3 times 2x minus 3. so you have to foil again 2x times 2x is 4x squared two x times negative three that's negative six x negative three times two x is also negative six x and finally negative three times negative three two negatives will change into a positive now just like before we need to combine like terms 6 plus 6 is 12 so negative 6 plus negative 6 is negative 12. so this is the answer 4x squared minus 12x plus 9. now let's say if we wish to multiply a binomial which has two terms with a trinomial which has three terms so initially after we multiply we should get a total of two times three terms which is six terms so let's multiply the five x by two x squared five times two is ten x times x squared is x cubed because one plus two is three so now let's multiply five x times negative three x five times negative 3 is negative 15 x times x is x squared next let's multiply 5x times 4. 5x times 4 is 20x and negative 9 times 2x squared that's negative 18x squared negative 9 times negative 3x that's positive 27x and we have one more negative nine times positive four that's negative 36 so now let's add like terms and let's arrange it so we have 10x cubed we can combine negative 15 and negative 18 that's negative 33 x squared and we combine these two 20x plus 27x that's 47x and there's nothing to combine the 36 with so we're just going to leave it as negative 36. so this is the answer to this particular problem now let's review some properties of exponents what's x cubed times x to the fourth power whenever you multiply common variables you need to add the exponents three plus four is seven so this is x to the seventh now what is x to the nine divided by x to the four whenever you're dividing you need to subtract the exponents nine minus four is five now what about when you raise one exponent to another exponent what do you need to do you need to multiply this is 7 times 6 which is 42. now let's explain why these rules work the way they do so let's start with multiplication we said that x squared times x cubed is x to the fifth power keep in mind x squared is simply x times x x cubed is x times x times x there's three x's multiplied so we have a total of five x's multiplied that's why when you multiply common variables you need to add the exponents as you can see we have five x's in this expression now why is it that when we raise one exponent to another we need to multiply instead of add it's two times three not two plus three why is it x to the sixth x squared to the third power means that we have three x squared and x squared each is x times x so we have a total of six x variables multiplied to each other that's why you need to multiply one exponent by the other whenever you raise an exponent by another one now what about when dividing let's say if we have x to the fifth power divided by x squared we said it's going to be five minus two which is three so this is x cubed another way in which you can see it is that x to the fifth power is equal to x times x times x times x times x it's five x's multiplied to each other x squared is simply x times x so you can cancel two x's on top and on the bottom and you're left over with three axes which is therefore x to the third so it's the same as doing five minus two which is three so now what about this example let's say if we have x to the fourth divided by x to the seventh what is the answer so four minus seven top minus bottom is negative three and whenever you have a negative exponent if you move it to the bottom that is if you move the variable to the bottom the negative exponent will become positive so x to the negative three is the same as one over x cubed and typically in algebra you want your final answer to contain positive exponents not negative exponents so this is the answer but now let's see if we can understand it x to the fourth we know it's x times x times x times x x to the seventh is x times x times x seven times so we can get rid of four x variables therefore we have three left over on the bottom and there's no more on top so three x's together is x cubed and thus we can see why this is the answer so now let's work on some practice problems what is three x to the fourth y to the fifth multiplied by five x to the sixth y to the seventh so first we need to multiply the three and the five three times five is fifteen next we need to multiply x to the fourth times x to six that's going to be four plus six which is ten and then we need to multiply y to the fifth times y to the seven five plus seven is twelve so the answer is fifteen x to the ten y to the twelve try this example what is eight x cubed y to the negative two multiplied by seven x to the negative eight y to the fifth so first we need to multiply eight and seven eight times seven is fifty-six x to the third times x to the negative eight three plus negative eight is negative five so it's x to the negative five and then y to the negative two times y to the five negative two plus five is three so we have y cubed notice that we have a negative exponent and we need to make it positive therefore we need to move x from the top to the bottom so the answer is going to be 56 y cubed divided by x to the fifth now what if we have a problem that looks like this 24 x to the seven y to the negative two divided by six x to the fourth y to the fifth what's the answer so we need to divide let's divide 24 by six 24 divided by six is four now what's x to the seven divided by x to the four we need to subtract top minus bottom seven minus four is three after you subtract it initially it goes on top now what about the next one y to the negative two divided by y to the fifth power so negative two minus five to subtract it you can use the number line so here's zero here's negative two so you need to go five units to the left whenever you subtract move to the left of the number line if you need to add move to the right so one two three four five this is negative three negative four negative five negative six negative seven so negative two minus five is negative seven and initially it goes on top so now to make the negative exponent positive we need to move the y to the bottom so the final answer is four x cubed divided by y to the seventh try this one 32 x to the fifth power y to the negative three z to the fourth divided by 40 x to the negative 8 y to the negative 7 z to the negative 8. so we can't divide 30 by i mean 32 by 40 nicely however notice that 8 can go into 32 and 40. so let's divide the top number and the bottom number by 8. 32 divided by 8 is 4. 40 divided by 8 is 5. so now we could focus on the variables x to the fifth power divided by x to the negative 8. so 5 minus negative eight five minus negative eight is the same as five plus eight which is thirteen so initially it's going to go on top so we have x to the thirteen on top now what's y to the negative three divided by y to the negative seven so what's negative three minus negative seven negative three minus negative seven is the same as negative three plus seven which is four so we have y to the fourth power now for the z's it's going to be 4 minus negative 8 which is the same as 4 plus 8 so that's 12. so this is the answer we don't have any negative exponents so we can leave the answer like this try this one three x cubed raised to the second power what is it equal so keep in mind whenever you raise one exponent to another you need to multiply so one times two is two and three times two that's six so we have three squared x to the sixth three squared is the same as three times three three times three is nine so the final answer is nine x to the sixth by the way what is the difference between five x squared and five plus x squared are these the same 5x squared is basically 25 x squared but 5 plus x squared is not 5 squared plus x squared it doesn't work that way instead we need to foil this is 5 plus x times 5 plus x which we've covered already so notice that the 5 and x are multiplied therefore you could distribute the two to the five and the x but here we have a plus sign in between whenever you see that you need to expand it like this and foil so five plus x times five plus x is going to be different five times five is 25 5 times x that's 5x x times 5 is also 5x and finally x times x is x squared so this is equal to 25 plus 10x plus x squared let's try another example try this one what is four x squared y to the third raised to the third power so four is the same as four to the first power so let's distribute the three three times one is three three times two is six and three times three is nine so what's four to the third power four to the third power is four times four times four four times four is sixteen and sixteen times four is sixty-four so this is the answer to that expression now what if you get a question that looks like this let's say it's eight x squared y to the fifth z to the sixth raised to the zero power anything raised to the zero power is always one it doesn't matter what's inside now let's say if you have negative two five x y cubed raised to zero what's the answer now so everything inside the parentheses is going to turn into a one so only this portion will become a one but notice that you have a negative two on the outside negative two times one is negative two so that's the answer for an expression that looks like that now let's say if you get an expression that looks like this five x to the negative two divided by y to the negative three times eight x to the four divided by y to the negative five what's the answer to this particular problem now the first thing i would do is try to get rid of the negative exponents so x to the negative 2 we can move it to the bottom it's going to become x squared and the y we can move it to the top and the same is true for the y to the negative five if we move it to the top it's going to be y to the positive five so now let's multiply across eight times five is forty and y cubed times y to the fifth we need to add the exponents it's going to be y to the eighth since three plus five is eight and for the x variables we need to divide x to the four divided by x squared that's four minus two so it's simply two so the final answer is forty x squared y to the eighth let's try this one 35 x to the negative 3 divided by 40 x y to the fifth power times 24 x squared y squared divided by 42 y to the negative four so how can we simplify this expression what do you think we need to do well what you don't want to do is you don't want to multiply across if you multiply 35 and 24 you're going to get a large number and if you multiply 40 and 42 you're going to get a bigger number and then you have to reduce the fraction it's a lot of work instead we need to simplify or break down these numbers into smaller numbers we can break down 35 into seven and five and forty we can make that eight times five now the x to negative three i'm going to move it to the bottom so i have x times x to the third on the bottom and y to the fifth twenty four i'm going to rewrite that as eight times three and forty two is seven times six but six i'm going to write that as three times two now y to the negative four i'm gonna move it to the top to make it y to the positive four so now let's see what we can cancel we can cancel on eight we can get rid of a 3 and we can get rid of a 7 and a 5. so on top we have an x squared y squared times y to the fourth two plus four is six so it's y to the six on top on the bottom we have a two we have x times x to the third which is x to the fourth and we also have y to the fifth so now two minus four is negative two but if we subtract it backwards four minus two is two and since the number on the bottom is bigger we're going to have x squared left over on the bottom if you do 2 minus 4 you're going to get negative 2 on top and then you'll have to move it to the bottom which will make it positive 2. for the y's we have 6 minus five which is one and that's going to stay on top so this is the final answer for this particular problem so now it's your turn try this one 24 x y divided by 27 x to the negative 2 divided by 36 x squared y to the negative 3 over 45 x y to the fourth now how can we divide these two expressions there's something called keep change flip you need to keep the first fraction the same way and then you could change division to multiplication if you flip the second fraction so now we can solve it the same way as we did in the last example so let's rewrite 24 as eight times three 27 we can make it nine times three and we can move the x to the negative two to the top forty five let's make it nine times five and thirty six we could make that nine times four or six times six so let's change 24 instead of making it eight times three let's make it six times four and 36 is going to be six times six and so we have x squared y to the negative three i'm going to move that to the top so now let's cancel a six we can cancel a nine and that's about it well actually the four we can change that into two times two and we have a five we have three on the bottom six we can make that three times two and that was this 6 that remained so now let's multiply the variables that we have on top so we can also cancel an x squared by the way so we're left over with x times x on top which is x squared and we also have y times y to the fourth times y to the third so one plus three plus four is eight so we have y to the eight on top so we can cancel a two so we have left over is two times five which is ten x squared y to the eight and on the bottom we have three times three which is nine so this is the final answer now consider the expression three x over five divided by seven x y over nine how can you simplify this expression so what we have here is a complex fraction we have a fraction within a fraction you can rewrite this as three x over five divided by seven x y over nine that's what it really means and so using keep change flip we can keep the first fraction the same change division to multiplication and flip the second fraction so now we can figure out the answer so we can cancel x and then we have to multiply across 3 times 9 is 27 5 times 7 is 35 so this is the answer 27 over 35 y now let's say if you have a complex fraction that looks like this 7 plus 2 over x divided by 5 minus 3 over y how would you simplify this expression the best way to simplify is to multiply top and bottom by the common denominator of these two fractions the common denominator is x and y so let's multiply the top part by x y and the bottom part by x y so x y times 7 is simply 7xy xy times 2 over x the x will cancel and so what we have left over is 2 times y now for the bottom part x times y times five is five x y and then x y times three over y y cancels so it's three x but negative three x now if we want to we can factor a y if you take out a y on top it's going to be 7 x plus 2 and on the bottom you can factor out an x if you take that out it's going to be 5 y minus 3 and so that's the answer so now we're going to focus on solving equations so let's say if we have the equation x plus 4 is equal to 9. how would you solve for x x is simply a variable it's an unknown number you want to find out what number plus 4 is 9. you know intuitively that five plus four is nine so x is equal to five but let's solve it to solve for x you need to get x on one side of the equation which means that you need to move the four to the other side the opposite of addition is subtraction so you need to subtract both sides by 4 to solve for x 4 and negative 4 cancels and so 9 minus 4 is 5. and that's how you can solve for x let's try another example what is three x plus five equal to eleven what is the value of x the first thing we need to do is subtract five from both sides eleven minus five is six so now how can we separate the three from the x notice that the three is multiplied to the x the opposite of multiplication is division so we'll need to divide both sides by three six divided by three is two so that is the value of x and to check it let's plug in two into the equation three times two plus five we know that three times two is six six plus five is eleven and so two is the correct answer for x let's try this one two times x minus one plus six is equal to ten so what's the first thing that we need to do to solve for x the first thing i would do is subtract six from both sides so 10 minus 6 is equal to 4. now to get rid of the parentheses we can either distribute 2 to x minus 1 or we could divide both sides by 2. and so on the left side since we no longer have a 2 in front of the parentheses we can get rid of the parentheses and so it's just x minus 1 on the left side on the right side 4 divided by 2 is 2. so to isolate x we just got to add 1 to both sides 2 plus 1 is 3 so therefore x is equal to 3. try this one five minus three times x plus four equals seven plus two times x minus one so in this particular example let's distribute the negative three and the two first so negative three times x is simply negative three x and negative three times four is negative 12. now let's distribute the 2 on the right side 2 times x is 2x and 2 times negative 1 is negative 2. so now let's add like terms 5 and negative twelve five minus twelve is negative seven on the right side we can combine seven and negative two seven minus two is five so now let's add 3x to both sides and simultaneously let's subtract 5 from both sides we need to do this in such a way that we can get x on one side of the equation so two x and three x they add to five x negative seven plus negative five is negative twelve so therefore five x is equal to negative twelve to separate the five from the x we need to divide both sides by five five divided by five is one so x is equal to negative 12 over 5 as an improper fraction here's another one what if we have 2 over 3 x plus 5 is equal to eight how would you solve for x the first thing i would do is subtract both sides by five so therefore two over three x is equal to eight minus five which is three now what we could do is multiply both sides by three on the left side the threes will cancel so all you have on the left side is two x on the right side three times three is nine so to separate the two from the x we need to divide both sides by two so the final answer is x is equal to nine over two which is about four point five let's try this one three over four x minus one third is equal to one how can we solve for x in this particular example when you have many fractions it's going to help a lot if you attempt to clear away all the fractions let's multiply both sides of the equation by the common denominator the common denominator of 4 and 3 is 12. so let's distribute 12 to each term so what's 3 4 x times 12 you can do 3 times 12 divided by 4 or you can do 12 divided by 4 times 3. 3 times 12 is 36 36 over 4 is 9. but if you do 12 divided by 4 that's 3 times the 3 on top you still get 9. either way it's going to be 9x now what's 12 times 1 third twelve times the third is the same as twelve divided by three which is four and then twelve times one is twelve so let's add four to both sides 12 plus 4 is 16 and if we divide both sides by 9 we can see that x is equal to 16 over 9. try this one x plus two over five is equal to seven over eight so how can we solve for x if we're given two fractions separated by an equal sign if you get a problem like this you can cross multiply five times seven is thirty five and 8 times x minus 2 you need to distribute the 8. it's going to be 8x plus 16. so to solve for x let's subtract both sides by 16. so 35 minus 16. let's subtract it 5 minus 6 is a negative number so let's borrow a 1. so 15 minus 6 is 9 2 minus 1 is 1. so thirty five minus sixteen is nineteen so what we now have is eight x is equal to nineteen so let's divide both sides by eight so x is nineteen over eight if you want to convert it to a mixed fraction need to realize that eight goes into nineteen two times eight times two is sixteen and nineteen minus sixteen is three so it's two and three eighths as a mixed number what if you have an equation that contains numbers how would you solve for x notice that most of the numbers that we have here there's two digits after the decimal point so therefore we need to multiply both sides by a hundred if there was only one digit after the decimal point would multiply both sides by ten so we got to multiply every number by a hundred point zero four x times 100 you simply need to move the decimal 2 units to the right so it's going to be 4x point 15 times 100 is 15 0.09 x times 100 is 9x 0.25 times 100 is 25 so now let's subtract 4x from both sides and simultaneously let's add 25 to both sides so these will cancel 15 plus 25 is 40. 9x minus 4x is 5x so to isolate x we need to divide both sides by 5. so 40 divided by 5 is 8. so that is the value of x for this particular problem and as you can see it's not that bad consider this equation how can we solve for x there's two ways in which we could solve for x the first thing we could do is we can add 25 to both sides so x squared is equal to 25 at this point we could take the square root of both sides the square root of x squared is simply x and the square root of 25 is plus or minus 5. so x can equal 5 or x can equal negative five five times five is twenty-five and negative five times negative five is also twenty-five so that's one way in which you can solve an equation that looks like that another technique that you can use is you can factor this expression using the difference of squares method the square root of x squared is x the square root of 25 is 5. on one side you're going to have a positive sign and on the other side you're going to have a negative sign so factoring is the opposite of foiling now you need to set each factor equal to zero so for the first one if you subtract both sides by five you'll see that x is equal to negative five and for the second expression if you add both sides by five you'll see that x can also be equal to positive five try this one two x squared minus eighteen is equal to zero how can we solve for x so we can't square root 2 and 18 we won't get a nice number however we can factor out the gcf the greatest common factor we can take out a 2 from 2x squared and 18. to find out what goes inside divide 2x squared divided by 2 is x squared negative eighteen divided by two is negative nine now notice that we can factor x squared minus nine using the difference of squares technique the square root of x squared is x the square root of nine is three and so it's going to be plus and minus therefore x can equal negative three and positive three let's try this one three x squared minus 48 equals zero feel free to pause the video and try this example it's very similar to the last two so the first thing we need to do is we need to remove the gcf which is three three can go into three x squared and forty eight three x squared divided by three is x squared and negative forty eight divided by three is negative sixteen which we can factor using the difference of squares technique the square root of x squared is x the square root of 16 is four and so it's going to be x plus four and x minus four so therefore x will equal negative four and positive 4. now let's say if you have an expression that looks like this x to the 4 minus 81 is equal to zero what is the value of x so notice that we can use the difference of squares technique the square root of x to the four is x squared because x squared times x squared is x to the fourth the square root of 81 is nine so one side is going to be plus and the other side is going to be minus now we can't factor a sum of squares but we can factor x squared minus nine because that's still a difference of squares and that's going to be x plus 3 times x minus 3. so therefore the real solutions that we have for x is negative 3 and positive 3. by the way the factor x squared plus nine can never be zero if you subtract both sides by nine you'll see that x squared is negative nine which can't be if you plug in three three squared is positive nine if you plug in negative three negative three times negative three is positive nine so you can't take the square root of a negative number the square root of negative nine is not a real solution it's an imaginary solution this is equal to 3i where i is equal to negative 1. so you won't get a real answer for x if you want to look for an imaginary answer then it's equal to plus or minus 3i now let's say if we have a trinomial x squared minus five x plus six is equal to zero how can we factor this expression to solve for x so notice that the leading coefficient is one what you need to do is find two numbers that multiply to six but that add to the middle term negative five so one and six won't work two and three is very close two times three is six but two plus three is positive five but negative two and negative three still multiplies to positive six but add to negative five so to factor it's going to be x minus two times x minus three so if you set each factor equal to zero we could solve for x here we need to add two to both sides so we can see that x is equal to positive two and for this one we need to add three to both sides so x is equal to positive three try this one x squared minus two x minus fifteen solve for x so what two numbers multiply to negative fifteen but add to the middle number negative two so it's not one and fifteen 3 and negative 5 3 times negative 5 is negative 15 but 3 plus negative 5 is negative 2. so it's going to be x plus 3 times x minus 5. therefore x is equal to negative 3. and positive five try this one x squared plus three x minus twenty eight is equal to zero so what two numbers multiply to negative 28 but add to three so we have two and fourteen and four and seven four and negative seven adds up to negative three but negative four and positive 7 adds up to positive 3 but still multiplies to negative 28 so it's going to be x minus 4 times x plus 7 which means that x is equal to positive 4. you need to change the sign and x is equal to negative seven so that's the solution to the equation now what if you have a trinomial where the leading coefficient is not one how can you factor this expression in order to solve for x in this case you need to multiply two and negative two two times negative two is negative four you need to find two numbers that multiply to negative four but still add to the middle term three so this is going to be four and negative one four plus negative one is positive three but four times negative one is still negative four now what we're going to do is we're going to replace the middle term 3x with 4x and negative 1x notice that the value of the expression is still the same 4x minus 1x is 3x now our next step is to factor by grouping so let's factor the gcf from the first two terms and the last two terms in the first two terms we can take out a 2x 2x squared divided by 2x is x and 4x divided by 2x is 2. in the last two terms let's take out a negative 1. negative 1x divided by negative 1 is simply positive x and negative two divided by negative one well that's going to be positive two so notice that we have a common factor x plus two when you see that that means you're on the right track if they're different you need to go back and check your work because it's an error somewhere so now we're going to factor out x plus 2. so what's going to be on the inside of the second parenthesis is the stuff on the outside that's the 2x and the negative one so now at this point we can factor the expression i mean that factor but solve it let's set each factor equal to zero and let's solve for x so for this one all we need to do is subtract both sides by two and so x is equal to negative two now for the other one we gotta do a little bit more work we need to add one to both sides so two x is equal to one and then we need to divide both sides by two so x is equal to one half so x is equal to negative two and x is equal to a half now sometimes if you're having difficulty factoring an expression you can always use the quadratic equation so let's use the quadratic equation for the example that we just worked on now this equation is called a quadratic equation and it's in standard form that's ax plus i mean ax squared plus bx plus c so therefore you could see that a is 2 b is 3 and c is negative 2. so using the quadratic equation it's negative b plus or minus the square root of b squared minus 4ac divided by 2a so it's going to be negative 3 because b is 3 plus or minus b squared or 3 squared minus four times a which is two times c which is negative two divided by two a or two times two so we have negative three plus or minus three squared is nine negative four times two is negative eight and negative eight times negative two is sixteen on the bottom two times two is four and 9 plus 16 is 25 and the square root of 25 is 5. so right now we could separate this into two equations negative three plus five divided by four and negative three minus five over four because we have a plus or minus negative three plus five that's two so it's two over four and negative 3 minus 5 is negative 8. 2 over 4 is the same as 1 half which is the first answer that we have negative 8 over 4 is negative 2 that's the second answer so if you're having difficulty solving it you can always use the quadratic equation to get the answer let's try another problem like the last example six x squared plus seven x minus three is equal to zero so solve this quadratic equation by factoring and by using the quadratic formula so the leading coefficient is not one so we need to multiply six and negative three six times negative three is negative eighteen now what two numbers multiply to negative eighteen but add to the middle term seven this is nine and negative two nine times negative two is negative eighteen but nine plus negative 2 is 7. so we're going to replace 7x with 9x and negative 2x it doesn't matter the order in which you put the two numbers you'll still get the same answer so now let's factor by grouping in the first two terms take out the gcf 3x can go into 6x squared and 9x so let's remove 3x 6x squared divided by 3x is 2x and nine x divided by three x is simply three and the last two terms let's take out a negative one negative two x divided by negative one is two x negative three divided by negative one is three so we have a common factor of 2x plus 3. so let's take that out and within the other parentheses it's going to be the 3x and the minus 1. so now let's set each factor equal to zero so two x plus three is equal to zero and three x minus one is equal to zero so here let's subtract both sides by three so two x is equal to negative three next let's divide both sides by two so therefore x is negative three over two now for the next example let's add one to both sides and so we could see that three x is equal to one and then let's divide both sides by three so x is one third so these are the answers positive one-third and negative three over two so now let's use the quadratic equation to get the same answer so x is equal to negative b plus or minus square root b squared minus 4ac over 2a so the first number in front of x squared this is a a six b is seven c is negative three so negative seven plus or minus b squared or seven squared minus four times six times negative three divided by two a or two times six seven squared is forty nine and 6 times 4 is 24 24 times 3 20 times 3 is 60 4 times 3 is 12 60 and 12 is 72 so 24 and 3 is 72 and there's two negative signs so it's going to be positive 72 2 times 6 is 12. so now what's 49 plus 72 so if we do it by hand 2 plus 9 is 11 carry over the 1. 4 and 1 is 5 plus 7 that's twelve so we have one twenty one inside the radical the square root of one twenty one is eleven so it's negative seven plus or minus 11 divided by 12. so we can separate it into two fractions negative 7 plus 11 over 12 and negative 7 minus eleven over twelve negative seven plus eleven that's four negative seven minus eleven is negative eighteen now four over twelve we can divide both numbers by four and so we can reduce it to one over three now for the next one we could divide both numbers by six eighteen divided by six is three twelve divided by six is two so we get the answers that we had in the last example so it's negative three over two and one third consider this expression x cubed minus four x squared minus x plus four is equal to zero how would you solve for x notice that the first two terms has the same ratio as the last two terms one and negative four and negative one and four whenever you see that and if you have a total of four terms you can factor by grouping so in the first two terms let's take out an x squared x cubed divided by x squared is x negative 4x squared divided by x squared is negative four and the last two terms let's take out a negative one negative x divided by negative one is x positive four divided by negative one is negative four so we have a common factor of x minus four now what we have left over on the outside is x squared minus one which we can factor that further using the difference of squares method so it's going to be x plus one times x minus one so the solutions are positive four negative one and positive one so we have three answers for cubic function now the next topic that we're going to go over is graphing linear equations so how can we graph the equation y is equal to two x minus one you need to realize that this is in slope intercept form which is mx plus b so m represents the number in front of x so m is equal to two m represents the slope so the slope is two that's rise over run b is the y intercept that's where it touches the y axis b is negative one so with this information we can graph the function so the first thing we should do is plot the y-intercept which is negative 1 on the y-axis so this is the y-axis this is the x-axis next we can use the slope to find the next point we said the slope is two or two over one which is rise over run so we need to rise two units up and then travel or run one unit to the right so therefore the next point is going to be at one comma one so then we're going to rise to run one so the next point is going to be over here now at this point we can draw a straight line and so that's how you can graph an equation in slope intercept form so let's try another example go ahead and graph y is equal to three over four x minus two so we could see that the y-intercept which is b is negative two so it's going to be on the y-axis and the slope which is the number in front of x is three fourths so that means we need to travel or go up three units and then four units to the right the rise is three the run is four so the next point is going to be four comma one and so we could connect these two points with a straight line and that line wasn't straight so let's try that again and so that's how you can graph it now sometimes you might have an equation in standard form standard form is ax plus b y is equal to c now in this form if you wish to graph it the best thing to do is to find the x and y-intercept to find the x-intercept substitute 0 for y so this completely disappears so two x is equal to six six divided by two is three so x is three so therefore the x intercept is three comma zero next let's find the y intercept to find the y intercept substitute zero for x so therefore this disappears so negative three y is equal to six six divided by negative three is negative two so we have the y intercept of zero negative two now we can plot those two points and connect them with a straight line so three zero is over here and zero negative two is over here and then just connect them so that's how you can graph an equation in standard form now sometimes you may need to write the equation of the line given a point and a slope so this three forms this form is the slope intercept form this is the standard form and this is the point-slope form so what we're going to do for each of these examples maybe not each of them but some of them we're going to start with the point-slope form convert it to the slope-intercept form and then convert that to the standard form so you know how to find all three if ever the need arises so let's say if the slope is two and you have the point one comma three how would you write the equation of the line that passed through the point one comma three and that has a slope of two so personally i think it's easier if you start with the point slope equation so this is x one and this is y one so let's replace y one with three m with two and x one with one so this is the answer in point slope form this is it it's y minus three is equal to two times x minus one now if you want to convert it to the slope intercept form distribute the two two times x is two x and two times negative one is negative two now let's add 3 to both sides the goal is to solve for y whenever you want it in point slope form not point slope form but slope intercept form if you want it in slope intercept form isolate y so y is equal to two x plus one so this is the answer in slope intercept form and this is the answer in point slope form now to convert it to standard form we simply need to get x and y on the left side of the equation so let's subtract both sides by 2x so therefore the equation in standard form is negative two x plus y is equal to one so now it's in ax plus b y equals c form now how can we write the equation of the line if we're given two points let's say the point two comma four and negative one comma five how can we write the equation of the line in this case we need to find the slope which is y2 minus y1 over x2 minus x1 so this is going to be x1 y1 and this is x2 y2 so it's going to be 5 minus four divided by negative one minus two five minus four is one negative one minus two is negative three so therefore the slope is negative one third so now let's write the equation of the line first in point slope form so y minus y one is equal to m times x minus x one so y one is four we can use either point two four or negative one five i'm going to use 4. so y 1 is 4 m is negative 1 3 and x 1 is 2. so this is the answer in point slope form so now let's convert it to slope intercept form let's distribute the negative one-third it's going to be negative one-third x and then negative one-third times negative two negative one times negative two is two so it's positive two-thirds now we need to add four to both sides so it's going to be y is equal to negative one third x plus two over three plus four four is the same as four over one to add two thirds and four we need to get common denominators so let's multiply four by three over three so four times three is twelve so two thirds plus twelve thirds two and twelve is fourteen so this is the final answer in slope intercept form you can see that the slope is negative one-third and the y-intercept is 14 over three so now let's put this equation in standard form so to put it in standard form the first thing we need to do is get rid of the fractions so let's multiply both sides by three so three times y is three y and three times negative one third x the three's cancel so it's simply negative one x and three times 14 over 3 the 3's cancel so you just get 14. so now we need to move the x to the other side so let's add x to both sides so the equation in standard form is going to be x plus 3y is equal to 14. so now it's in ax plus b y equals c4 now sometimes you may need to write the equation of the line given another parallel or perpendicular line so let's say if you want to write the equation of the line that passes through the point one comma three and that's parallel to the equation 2x minus 3y minus 5 is equal to 0. how would you do it keep in mind to write the equation of any line you need the point which we already have and the slope so we could find a slope using this equation since it's parallel the slopes will be the same so let's put that equation in slope intercept form and the number in front of x in that form will be the slope so let's solve for y so starting with this equation let's subtract both sides by 2x and let's add 5 at the same time so on the left side we're going to have negative 3y and on the right side it's going to be negative 2x plus 5. so now all we need to do is divide by negative 3 to each term so the 3s are going to cancel on the left side so therefore y is positive two over three x minus five thirds so therefore we could see that the slope which is the number in front of x is two thirds so now we can write the equation of the line let's write it in slope intercept form so instead of using the point slope equation we're going to use this one directly so we're going to substitute y for three we're gonna plug in two thirds for m and replace x with one and let's solve for b so let's multiply both sides by three to get rid of the fraction so three times three is nine and three times two thirds the three's cancel so you get two and don't forget to distribute the three to b so it's streaming so now let's subtract both sides by two so nine minus two is seven seven is equal to three b so now we can divide both sides by three to isolate b so b is equal to seven over three so now we can write the equation of the line in slope intercept form so y equals mx plus b all we need to do is replace m and b so m is 2 over 3 and b is seven over three so this is the equation of the line that is parallel to this line but passes through the point one three so here's the last question for the day the equation of the line that passes through the point negative two comma one and that's perpendicular to the equation three x plus two y minus seven is equal to zero so just like before first we need to convert this equation to standard form not standard form but slope intercept form once it's in slope intercept form we can find the slope and then we could find the slope of the perpendicular line so let's solve for y so let's move the 3x and the 7 to the other side let's subtract both sides by 3x and let's add 7 to both sides so two y is equal to negative three x plus seven and now let's divide each number by two so y is equal to negative three over two x plus seven over two so the slope of this line is negative three over two therefore the slope of the perpendicular line it's going to be the negative reciprocal of that value so it's going to be positive 2 over 3. you need to change the negative sign to a positive sign and flip the fraction at the same time so now that we have the point and the slope we can write the equation so let's use the slope-intercept form equation so let's replace y with one m with two thirds and x with negative two now to get rid of the fraction let's multiply everything by three so three times one that's three three times two thirds times negative two the threes will cancel and so what we have left over is 2 times negative 2 which is just negative 4 and then 3 times b which is 3b so now let's add 4 to both sides so therefore 7 is equal to three b and now let's divide both sides by three so b is seven over three so now we can write the equation so let's plug in the values for m and b so it's going to be two over three x plus seven over three so this is the equation of the line you