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 3 radical 2 plus 5 radical 7 plus 8 radical 2 plus 3 radical 7. These are like terms. Because the radical 2 is the same, You can add the 3 and 5, not the 3 and 5, but the 3 and 8. 3 plus 8 is 11, so it's 11 root 2. Now, these two are like terms.
You can add the 5 and the 3 to give you 8, so it's 8 root 7. Try this one. 7x plus 4x squared plus 5x. plus 9x squared. So we can add the 4x squared and the 9x squared because they're like terms.
They both have an x squared. 4 plus 9 is 13x squared. We can add the 7x and the 5x. 7 and 5 is 12. So this is the answer. Try this one.
Let's say if you have 9x squared plus 6x plus 5 plus 3x squared minus 5x minus 9. So notice that we can add 9x squared and 3x squared. They're like terms, so that's going to be 12x squared. Now, we can also add 6x and 5x. 6x plus negative 5x is the same as 6x minus 5x.
6 minus 5 is 1, so we're just going to get 1x. 5 plus negative 9, 5 minus 9 is negative 4. The final answer is 12x squared plus x minus 4. Here's another example you can try. 3x squared plus 7x minus 4 minus 8x squared minus 5x plus 7. 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 3x squared plus 7x minus 4 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 it's going to be negative 8x squared and instead of having negative 5x it's going to be positive 5x And instead of having positive 7, if we distribute the negative sign to it, it's negative 7. And so now, let's add like terms. 3x squared plus negative 8x squared.
3 minus 8 is negative 5, so it's negative 5x squared. 7x plus 5x. 7 plus 5 is 12, so it's 12x. Negative 4 plus negative 7 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 3x squared y is a monomial. A binomial has two terms, like 5x plus 6, 7x minus 3, those are binomials. A trinomial has three terms. So x squared plus 6x plus 5 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. 1 plus 2 is 3, 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 3 X's multiplied to each other.
That's why it's X cubed. Therefore, 7x times x squared is 7x cubed. Now what is 7x times 2x?
7 times 2 is 14. x times x, or x to the first power times x to the first power. 1 plus 1 is 2, so it's x squared. And then 7x times negative 3 is negative 21x.
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 4th power minus 6x cubed plus 5x minus 8. So let's distribute the 5x squared. 5x squared times 3x to the fourth 5 times 3 is 15 X squared times X to the fourth is X to the 6 you need to add 2 and 4 2 plus 4 is 6 So now we need to multiply 5x squared times negative 6x cubed. 5 times negative 6 is negative 30. x squared times x cubed, 2 plus 3 is 5. So it's going to be x to the 5th power.
And then 5x squared times 5x. 5 times 5 is 25. x squared times x, and we have 2 plus 1, which is 3, and so it's simply going to be x cubed. And then 5x squared times negative 8, that's 5 times negative 8 is negative 40, so it's negative 40x 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 2x plus 7? So what we need to do here is something called... foil.
We need to foil these two expressions. What's 3x times 2x? 3 times 2 is 6, and x times x is x squared. Keep in mind, this is x to the first power.
1 plus 1 is 2. So next we need to multiply the 3x by the 7. 3x 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 FOIA 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 20x. 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.
2x times negative 3, that's negative 6x. Negative 3 times 2x is 4x squared. is also negative 6x and finally negative 3 times negative 3 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 for x 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 it, we should get a total of... 2 times 3 terms which is 6 terms.
So let's multiply the 5x by 2x squared. 5 times 2 is 10. X times x squared is x cubed because 1 plus 2 is 3. So now let's multiply 5x times negative 3x. 5 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 9 times positive 4. 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 33x 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. 3 plus 4 is 7, so this is x to the 7. Now what is x to the 9 divided by x to the 4?
Whenever you're dividing, you need to subtract the exponents. 9 minus 4 is 5. 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. this expression now why is it that when we raise one exponent to another we need to multiply instead of add it's 2 times 3 not 2 plus 3 why is it X to the 6 x squared to the third power means that we have three x squareds 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 5 minus 2 which is 3 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 X's, 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 4th divided by x to the 7th. What is the answer?
So 4 minus 7, top minus bottom, is negative 3. 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 3 is the same as 1 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 4th, we know it's x times x times x times x.
X to the 7th is x times x times x, 7 times. So we can get rid of 4x variables. Therefore, we have 3 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 3x to the 4th, y to the 5th, multiplied by 5x to the 6th, y to the 7th? So first, we need to multiply the 3 and the 5. 3 times 5. is 15. Next, we need to multiply x to the fourth times x to the sixth. That's going to be 4 plus 6, which is 10. And then we need to multiply y to the fifth times y to the seventh. 5 plus 7 is 12. So the answer is 15x to the 10y to the 12th. Try this example.
What is 8x cubed y to the negative 2 multiplied by 7, x to the negative 8, y to the 5th. So first, we need to multiply 8 and 7. 8 times 7 is 56. x to the 3rd times x to the negative 8. 3 plus negative 8 is negative 5, so it's x to the negative 5. And then, y to the negative 2 times y to the 5. Negative 2 plus 5. is 3, 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 56y cubed divided by x to the 5th. Now what if we have a problem that looks like this? 24. x to the 7, y to the negative 2, divided by 6, x to the 4, y to the 5th.
What's the answer? So, we need to divide. Let's divide 24 by 6. 24 divided by 6 is 4. Now, what's x to the 7 divided by x to the 4?
We need to subtract. Top minus bottom. 7 minus 4 is 3. After you subtract it, initially it goes on top.
Now what about the next one? y to the negative 2 divided by y to the 5th power. So negative 2 minus 5. To subtract it, you can use the number line. So here's 0, here's negative 2. So you need to go 5 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 1, 2, 3, 4, 5. This is negative 3, negative 4. Negative 5, negative 6, negative 7. So negative 2 minus 5 is negative 7. 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 4x cubed divided by y to the 7th.
try this one 32 X to the fifth power Y to the negative 3 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 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 can focus on the variables. x to the 5th power divided by x to the negative 8. So 5 minus negative 8. 5 minus negative 8 is the same as 5 plus 8, which is 13. So initially it's going to go on top, so we have x to the 13 on top.
Now what's y to the negative 3 divided by y to the negative 7? So what's negative... 3 minus negative 7 negative 3 minus negative 7 is the same as negative 3 plus 7 which is 4 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.
3x cubed raised to the second power. What is it equal? So keep in mind, whenever you raise 1x... point to another you need to multiply so 1 times 2 is 2 and 3 times 2 that's 6 so we have 3 squared X to the 6 3 squared is the same as 3 times 3 3 times 3 is 9 so the final answer is 9 X to the 6 By the way, what is the difference between 5x squared and 5 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 five 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 5 plus X times 5 plus X is going to be different 5 times 5 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 4x squared y to the third raised to the third power? So 4 is the same as 4 to the first power. So let's distribute the 3. 3 times 1 is 3. 3 times 2 is 6. 3 times 3 is 9. So what's 4 to the 3rd power?
4 to the 3rd power is 4 times 4 times 4. 4 times 4 is 16, and 16 times 4 is 64. So this is the answer to that expression. Now what if you get a question that looks like this? Let's say it's 8x squared y to the 5th z to the 6th raised to the 0 power.
Anything raised to the 0 power is always 1. It doesn't matter what's inside. Now let's say if you have negative 2 5xy cubed raised to the 0. What's the answer now? So everything inside the parentheses is 1. is going to turn into a 1. So only this portion will become a 1. But notice that you have a negative 2 on the outside. Negative 2 times 1 is negative 2. 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.
5x to the negative 2 divided by y to the negative 3 times 8x to the 4 divided by y to the negative 5. 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 5 if we move it to the top it's going to be Y to the positive 5 so now let's multiply across 8 times 5 is 40 and Y Q times y to the fifth we need to add the exponents it's going to be y to the 8 since 3 plus 5 is 8 and for the X variables we need to divide X to the 4 divided by x squared that's 4 minus 2 so it's simply 2 so the final answer is 40 x squared y to the 8 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 4. 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 7 and 5. And 40, we can make that 8 times 5. Now, the x to the negative 3, 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. 24, I'm going to be right there. that as 8 times 3 and 42 is 7 times 6 but 6 I'm going to write that as 3 times 2 now y to the negative 4 I'm going to move it to the top to make it y to the positive 4 so now let's see what we can cancel we can cancel an 8 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, 2 plus 4 is 6, so it's y to the 6 on top. On the bottom, we have a 2. We have x times x to the third, which is... X to the fourth and we also have Y to the fifth so now 2 minus 4 is negative 2 but if we subtract it backwards 4 minus 2 is 2, 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 5, which is 1, 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.
24xy. 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 8 times 3. 27, we can make it 9 times 3. And we can move the x to the negative 2 to the top. 45, let's make it 9 times 5. And 36, we could make that 9 times 4. or 6 times 6 so let's change 24 instead of making it 8 times 3 let's make it 6 times 4 and 36 is going to be 6 times 6 and so we have x squared y to the negative 3 I'm going to move that to the top So now, let's cancel a 6. We can cancel a 9. And that's about it.
Well, actually, the 4, we can change that into 2. times 2 and we have a 5 we have three in the bottom six we can make that three times two and that was this six that remained so now let's multiply the variable 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 1 plus 3 plus 4 is 8 so we have y to the eighth on top so we can cancel a 2 so what we have left over is 2 times 5 which is 10 x squared y to the 8 and on the bottom we have 3 times 3 which is 9 so this is the final answer Now, consider the expression 3x over 5 divided by 7xy over 9. How can you simplify this expression? So what we have here is a complex fraction. we have a fraction within the fraction you can rewrite this as 3x over 5 divided by 7xy over 9 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 35y. 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 xy and the bottom part by xy.
So, xy 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 5 is 5xy.
And then xy times 3 over y, and y cancels, so it's 3x, but negative 3x. Now if we want to, we can factor a y. If you take out a y on top, it's going to be 7x plus 2. And on the bottom, you can factor out an x. If you take that out, it's going to be 5y 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 5 plus 4 is 9. So x is equal to 5. 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 4. 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 3x plus 5 equal to 11? What is the value of x?
The first thing we need to do... is subtract 5 from both sides. 11 minus 5 is 6. So now how can we separate the 3 from the x? Notice that the 3 is multiplied to the x. The opposite of multiplication is division.
So we'll need to divide both sides by 3. 6 divided by 3 is 2. So that is the value of x. And to check it, let's plug in 2 into the equation. 3 times 2 plus 5. We know that 3 times 2 is 6. 6 plus 5 is 11. And so, 2 is the correct answer for x.
Let's try this one. 2 times x minus 1 plus 6 is equal to 10. So what's the first thing that we need to do to solve for x? The first thing I would do is subtract 6 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 gotta add 1 to both sides. 2 plus 1 is 3, so therefore x is equal to 3. Try this one. 5 minus 3 times x plus 4 equals 7 plus 2 times x minus 1. so in this particular example let's distribute the negative 3 and the two first so negative 3 times x is simply negative 3x and negative 3 times 4 is negative 12 now let's distribute the two 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 12 5 minus 12 is negative 7. On the right side we can combine 7 and negative 2. 7 minus 2 is 5. 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 2x and 3x, they add to 5x. Negative 7 plus negative 5 is negative 12. So therefore, 5x is equal to negative 12. To separate the 5 from the x, we need to divide both sides by 5. 5 divided by 5 is 1. So x is equal to negative 12 over 5, as an improper fraction.
Here's another one. What if we have 2 over 3x plus 5 is equal to 8? How would you solve for x? The first thing I would do is subtract both sides by 5. so therefore 2 over 3 X is equal to 8 minus 5 which is 3 now what we could do is multiply both sides by 3 on the left side the threes will cancel so I All you have on the left side is 2x. On the right side, 3 times 3 is 9. So to separate the 2 from the x, we need to divide both sides by 2. So the final answer is x is equal to 9 over 2, which is about 4.5.
Let's try this one. 3 over 4. x minus 1 third is equal to 1. 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 fourths 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?
12 times 1 third is the same as 12 divided by 3, which is 4. And then 12 times 1 is 12. So let's add 4 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 2 over 5 is equal to...
7 over 8. 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. 5 times 7 is 35 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 35 minus 16 is 19. So what we now have is 8x is equal to 19. So let's divide both sides by 8. So x is 19 over 8. If you want to convert it to a mixed fraction, you need to realize that 8 goes into 19 two times. 8 times 2 is 16, and 19 minus 16 is 3, so it's 2 and 3 eighths as a mixed number. What if you have an equation that contains decimal 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 100. If there was only one digit after the decimal point, we'd multiply both sides by 10. So we're going to multiply every number by 100. 0.04x times 100, you simply need to move the decimal two units to the right. So it's going to be 4x. 0.15 times 100 is 15. 0.09x...
times 100 is 9x point 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 can solve for x. The first thing we can do is we can add 25 to both sides.
So x squared is equal to 25. At this point, we can 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 5. 5 times 5 is 25, and negative 5 times negative 5 is also 25. 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 0. So for the first... one if you subtract both sides by 5 you'll see that X is equal to negative 5 and for the second expression if you add both sides by 5 you'll see that X can also be equal to positive 5 Try this one.
2x squared minus 18 is equal to 0. 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 two from 2x squared and 18 to find out what goes inside divide 2x squared divided by 2 is x squared Negative 18 divided by 2 is negative 9. Now notice that we can factor x squared minus 9 using the difference of squares technique.
The square root of x squared is x. The square root of 9 is 3. And so it's going to be plus and minus. Therefore, x can equal negative 3 and positive 3. Let's try this one.
3x squared minus 48 equals 0. 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 3. 3 can go into 3x squared and 48. 3x squared divided by 3 is x squared, and negative 48 divided by 3 is negative 16, which we can factor using the difference of squares technique. The square root of x squared is x, the square root of 16 is 4, and so it's going to be x plus 4 and x minus 4. So therefore, x will equal negative 4 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 0. What is the value of x? So notice that... we can use the difference of squares technique. The square root of x to the 4th is x squared, because x squared times x squared is x to the 4th. The square root of 81 is 9. 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 9 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 9 can never be 0. If you subtract both sides by 9, you'll see that x squared is negative 9, which can't be. If you plug in 3, 3 squared is positive 9. If you plug in negative 3, negative 3 times negative 3 is positive 9. So you can't take the square root of a negative number.
The square root of negative 9... 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 5x plus 6 is equal to 0. How can we factor this expression to solve for x?
So notice that the leading coefficient is 1. What you need to do is find two numbers that multiply to 6, but that add to the middle term negative 5. So 1 and 6 won't work. 2 and 3 is very close. but 2 plus 3 is positive 5. But negative 2 and negative 3 still multiplies to positive 6, but add to negative 5. So to factor it, it's going to be x minus 2 times x minus 3. So if you set each factor equal to 0, we could solve for x.
Here we need to add 2 to both sides. So we can see that x is equal to positive 2. And for this one, we need to add 3 to both sides. So x is equal to positive 2. positive 3. Try this one. x squared minus 2x minus 15. Solve for x.
So what two numbers multiply to negative 15 but add to the middle number negative 2 so it's not 1 in 15 3 and negative 5th coordinate 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 5 Try this one. x squared plus 3x minus 28 is equal to 0. So what two numbers multiply to negative 28 but add to 3? So we have 2 and 14. and 4 and 7. 4 and negative 7 adds up to negative 3, but negative 4 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 7. So, that's the solution to the equation. Now, what if you have a trinomial where the leading coefficient is not 1? How can you factor this expression in order to solve for x?
In this case, you need to multiply 2 and negative 2. negative 4 you need to find two numbers that multiply to negative 4 but still add to the middle term 3 so this is going to be 4 and negative 1 4 plus negative 1 is positive 3 but 4 times negative 1 is still negative 4 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 2 divided by negative 1, well, that's going to be positive 2. So, notice that we have a common factor, x plus 2. 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 parentheses is the stuff on the outside. That's the 2x and the negative 1. So now at this point, we can factor the expression. I mean, not factor, but solve it.
Let's set each factor equal to 0. And let's solve for x. So for this one, all we need to do is subtract both sides by 2. And so x is equal to negative 2. Now for the other one, we've got to do a little bit more work. We need to add 1 to both sides. So 2x is equal to 1. And then we need to divide both sides by 2. So x is equal to 1 half. So x is equal to negative 2 and x is equal to 1 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 squared plus bx plus c. So therefore, you can 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 4 times a which is 2 times c which is negative 2 divided by 2a or 2 times 2. So we have negative 3 plus or minus 3 squared is 9. Negative 4 times 2 is negative 8. And negative 8 times negative 2 is 16. On the bottom, 2 times 2 is 4. 9 plus 16 is 25 and the square root of 25 is 5 so right now we could separate this into two equations negative 3 plus 5 divided by 4 and negative 3 minus 5 over 4 because we have a plus minus negative 3 plus 5 that's 2 so it's 2 over 4 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. 6x squared plus 7x minus 3 is equal to 0. So solve this quadratic equation by factoring by using the quadratic formula so the leading coefficient is not one so we need to multiply 6 and negative 3 6 times negative 3 is negative 18 now what two numbers are multiplied to negative 18 but at to the middle term 7. This is 9 and negative 2. 9 times negative 2 is negative 18, but 9 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 9x divided by 3x is simply 3. In the last two terms, let's take out a negative 1. Negative 2x divided by negative 1 is 2x. Negative 3 divided by negative 1 is 3. 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.
x and the minus one. So now let's set each factor equal to zero. So 2x plus 3 is equal to zero and 3x minus one is equal to zero.
equal to 0 so here let's subtract both sides by 3 so 2x is equal to negative 3 next let's divide both sides by 2 so therefore X is negative 3 over 2 Now for the next example, let's add 1 to both sides. And so we can see that 3x is equal to 1. And then let's divide both sides by 3. So x is 1 third. So these are the answers.
Positive 1 third and negative 3 over 2. 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 is 6, b is 7, c is negative 3. So negative 7 plus or minus b squared or 7 squared minus 4 times 6 times negative 3 divided by 2a or 2 times 6. 7 squared is 49 and 6 times 4 is 24. 24 times 3. 20 times 3 is 60. 4 times 3 is 12. 60 and 20. 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 12. So we have 121 inside the radical. The square root of 121 is 11, so it's negative 7 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 11 over 12. Negative 7 plus 11, that's 4. Negative 7 minus 11 is negative 18. Now 4 over 12, we can divide both numbers by 4. And so we can reduce it to 1 over 3. Now for the next one, we can divide both numbers by 6. 18 divided by 6 is 3. 12 divided by 6 is 2. So we get the answers that we had in the last example. So it's negative 3 over 2 and 1 third.
Consider this expression. x cubed minus 4x squared minus x plus 4 is equal to 0. How would you solve for x? Notice that the first two terms has the same ratio as the last two terms. 1 in negative 4 and negative 1 in 4. Whenever you see that... that and if you have a total of four terms you can factor by grouping so in the first two terms let's take on x squared x cubed divided by x squared is x negative 4x squared divided by x squared is negative 4 and the last two terms let's take out a negative 1 negative x divided by negative 1 is x positive 4 divided by negative 1 is negative 4 so we have a common factor of X minus 4 now what we have left left over on the outside is x squared minus 1, which we can factor that further using the difference of squareness method.
So it's going to be x plus 1 times x minus 1. So the solutions are positive 4, negative 1, and positive 1. So we have three answers for a cubic function. Now the next topic that we're going to go over is graph and linear equations. So how can we graph the equation y is equal to 2x minus 1?
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 2. M represents the slope, so the slope is 2. That's rise over run. B is the y-intercept.
That's where it touches the y-axis. B is negative 1. 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 a slope to find the next point. We said the slope is 2, or 2 over 1. which is rise over run. So we need to rise 2 units up, and then travel or run 1 unit to the right. So therefore, the next point is going to be at 1,1.
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 and slope in a set form so let's try another example go ahead and graph y is equal to 3 over 4 X minus 2 So we can see that the y-intercept, which is b, is negative 2, so it's going to be on the y-axis. And the slope, which is the number in front of x, is 3 fourths. So that means we need to travel or go up 3 units and then 4 units to the right.
The rise is 3, the run is 4. So the next point is going to be 4, 1. And so we can 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 by 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 2x is equal to 6 6 divided by 2 is 3 so x is 3 so therefore the x-intercept is 3,0. Next, let's find the y-intercept.
To find the y-intercept, substitute 0 for x. So, therefore, this disappears. So, negative 3y is equal to 6. 6 divided by negative 3 is negative 2. So, we have the y-intercept of 0, negative 2. Now, we can plot those two points and connect them with a straight line.
So, 3, 0 is over here, and 0, negative 2 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, there's 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 2 and you have the point 1,3.
How would you write the equation of the line that passes through the point 1,3 and that has a slope of 2? So personally, I think it's easier if you start with the point slope equation. So this is x1 and this is y1. So let's replace y1 with 3, m with 2, and x1 with 1. So this is the answer in point-slope form. This is it.
It's y minus 3 is equal to 2 times x1. minus 1 now if you want to convert it to the slope intercept form distribute the 2 2 times X is 2x and 2 times negative 1 is negative 2 Now let's add 3 to both sides. The goal is to solve for y whenever you want it in slope-intercept form.
If you want it in slope-intercept form, isolate y. So y is equal to 2x plus 1. 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 2x plus y is equal to 1. So now it's in ax plus by equals c form. Now, how can we write the equation of the line if we're given two points? Let's say the point 2, 4 and negative 1, 5. How can we write the equation of the line?
In this case, we need to find the slope, which is y. 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 4 divided by negative 1 minus 2. 5 minus 4 is 1, negative 1 minus 2 is negative 3, so therefore the slope is negative 1 third. So now let's write the equation of the line first in point slope form. So y minus y1 is equal to m times x minus x1. So y1 is 4, we can use either point 2, 4, or negative 1, 5. I'm going to use 2, 4. So y1 is 4, m is negative 4. 1 3rd and x1 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 1 3rd it's going to be negative 1 3rd X and then negative 1 3rd times negative 2 negative 1 times negative 2 is 2 so it's positive 2 thirds Now, we need to add 4 to both sides.
So, it's going to be y is equal to negative 1 third x plus 2 over 3 plus 4. 4 is the same as 4 over 1. To add 2 thirds and 4, we need to get common denominators. So, let's multiply 4 by 3 over 3. so four times three is 12 so two-thirds plus 12 thirds 2 and 12 is 14 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 3 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 3. So 3 times y. is 3y and 3 times negative 1 third x the 3's cancel so it's simply negative 1x and 3 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 by equals c form.
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 1,3 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 3's are going to cancel on... left side so therefore y is positive 2 over 3x minus 5 thirds so therefore we could see that the slope which is the number in front of X is 2 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 3, and we're going to plug in 2 thirds for m, and replace x with 1. And let's solve for b. So let's multiply both sides by 3 to get rid of the fraction. Thank you.
So 3 times 3 is 9, and 3 times 2 thirds, the 3's cancel, so you get 2. And don't forget to distribute the 3 to B, so it's 3B. So now let's subtract both sides by 2. So 9 minus 2 is 7, 7 is equal to 3B. So now we can divide both sides by 3 to isolate B.
So B is equal to 7 over 3. 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 7 over 3. So this is the equation of the line that is parallel to this line but passes through the point 1, 3. So here's the last question for the day. Write the equation of the line that passes through the point negative 2, 1, and that's perpendicular to the equation 3x plus 2. 2y minus 7 is equal to 0 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 a slope and then we could find a 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 2y is equal to negative 3x plus 7. And now let's divide each number by 2. So y is equal to negative 3 over 2x plus 7 over 2. 2 so the slope of this line is negative 3 over 2 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 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 1, m with 2 thirds, and x with negative 2. Now, to get rid of the fraction, let's multiply everything by 3. So, 3 times 1, that's 3. 3 times 2 thirds times negative 2. The 3's 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. both sides so therefore 7 is equal to 3b and now let's divide both sides by 3 so B is 7 over 3 So now we can write the equation. So let's plug in the values for m and b.
So it's going to be 2 over 3x plus 7 over 3. So this is the equation of the line.