welcome to math tv with professor v this is video lecture for college algebra on section 1.2 quadratic equations so a quadratic equation is an equation equivalent to one of the form ax squared plus bx plus c equals zero where a b and c are real numbers and a cannot be equal to zero otherwise it wouldn't be quadratic so the difference from the equations we're going to be solving in this video versus the last lesson is that now our variable is going to be squared not just to the first power okay that's what makes it a quadratic equation so we're going to look at two ways that you can solve quadratic equations the first one is by factoring that's the easier method that one's more straightforward so you want to try factoring first and please keep this in mind note of caution always set the equation equal to zero before factoring because the whole point is you want to use the zero product property and you can't unless the equation is equal to zero okay yes so here we go example one says solve each equation and we're solving by factoring so we have to be obedient x squared minus 25 equals zero we're going to factor left hand side x squared minus 25 right hand side we've got a zero which is good so we can start thinking how do we factor x squared minus 25 well it's a difference of squares so it's gonna factor into conjugate pairs x plus five x minus five and then this is equal to zero the whole point of having a zero on the right hand side is for this next step here so if x plus 5 times x minus 5 is equal to 0 then by the zero product property either x plus 5 is 0 or x minus five is equal to zero or both you can't do that with any other number that's why it's called the zero product property the zero is essential and then from here you've got two cute little linear equations you just gotta solve them so subtracting five from the first one we have x equals negative five or i add five here x equals positive five okay we don't have any restrictions so those can both go in our solution set you can list them separately like this negative five five or if you're feeling like being efficient a little fancy you can just put plus or minus five that's cool too okay now you might be saying oh i know another way to solve the equation i move the 25 over i take the square root that's cool but the direction said to solve by factoring so we got to be obedient yes okay example b we have 3x squared plus 5x equals negative 2. no don't start factoring just yet remember we need a 0 on one side of the equation usually the right side of the equation is where i like to put the zeros so add 2 to both sides now we have three x squared plus five x plus two is equal to zero okay good now from here you can factor either by the ac method and splitting the middle term if you need to or i just do guess and check so i set it up i think to myself all right 3x squared the only way to get 3x squared is with three x and x yes good two only way to get a two is two times one so we just gotta figure out does the two go here and the one go here or is it the other way around so you just check just check in your head what works since i want 5x in the middle then i'm going to want yeah 3 times 1 3x times 1 and 2x right there that'll give me my 5x so we're good everything's positive plus plus okay good then from here set each factor equal to zero so three x plus two equals zero or x plus one equals zero so three x equals negative two x is negative two thirds or x equals negative one okay we didn't talk about restrictions but there aren't any so far so i'm not stressing i will start stressing in just a moment okay good so let's move on here now these are kind of reminding you of the equations in the last lesson so we have 5 over x plus 4 equals 4 plus 3 over x minus 2. so yeah to start off we do have restrictions remember we don't want our denominators to be zero so x plus four what would make it zero if x was negative four right so x cannot equal negative four this four here is not bothering me it has no variables in the denominator and then x minus two i don't want that denominator to be zero so x cannot equal positive two either so those are our restrictions okay now second step think back to the steps that we had in the last lesson the first two steps always apply we're going to multiply through by the lcd to get rid of fractions so the lcd is just going to be x plus 4 times x minus two okay and i'll write everything nice and spread out so you can multiply through by the lcd and see how everything cancels nicely okay three over x minus two so everybody gets multiplied by x plus four x minus two in the numerator remember and the goal is to be fraction free in the next step so let's see let's see x plus 4 cancels so i just have 5 times x minus 2 looking good so far equals nothing cancels here that's fine i mean there was no denominator so 4 times x plus 4 times x minus 2 plus what's going to cancel here the x minus 2 bam bam so i just have 3 times x plus 4. all right now we got to clean up distribute so this is going to be 5x minus 10 equals now please do this carefully when you have one little term out here a monomial and then you've got these two little binomials two factors to multiply you multiply this all out first then distribute the four after okay the little loner outside he's last i know so sad okay so we've got four times this is going to be x squared minus two x plus four x so plus 2x and then minus 8 like that plus and then just distribute 3x plus 12 okay good keep cleaning up so 5x minus 10 equals i'm going to distribute the 4 now so 4x squared plus 8x minus 32 yeah plus 3x plus 12. okay let's combine some like terms it's looking messy i know 5x minus 10 equals 4x squared then here we've got 8x plus 3x that is 11x then we have minus 32 plus 12 so that's minus 20. okay we need to set everything equal to zero before we can start factoring i it's not a good idea to factor with x squared having a negative coefficient so i'm just gonna move everything to the right because i don't want this to become negative by moving it to the left you get what i'm saying okay good so minus 5x plus 10 minus 5x plus 10. this is gone now i have zero equals four x squared plus six x minus ten okay before we start factoring i'm noticing i'm noticing i can take out a 2 from everything on the right hand side right 4 divides by 2 6 divides by 2 10 divides by 2. what i'm going to do to make life easy is divide the whole equation by two because guess what zero divided by two is still zero and then we have two x squared plus three x minus five that's gonna be so much nicer to factor and you'll still get the same solutions right so let's try to factor here 2x squared plus 3x minus 5 equals 0. so we have 2x and x i want three in the middle five is gonna be five times one so let's put the one here five here that should be positive okay if you're struggling with guess and check then your other option i'll write it up this way zero equals two x squared plus three x minus five remember we called the coefficient on x squared a this is b constant is c and then you can do the ac method and split the middle term so a times c goes in the top air conditioner usually is upstairs negative 10 that's not necessarily true just helps me remember okay and then b here's your basement 3. can you think of two numbers that multiply to negative 10 and add up to 3 so what multiplies to 10 1 and 10 that's not gonna add up to three two and five yes so since it's a negative ten and a positive three we would have a negative two and a positive five and then what do you do you split the middle term with those numbers so we have zero equals two x squared minus two x plus five x minus five right if i were to combine these i'd get back my 3x so haven't done anything illegal then when you have four terms you factor by grouping so for the first two terms i can take out a 2x and i have x minus 1 left and then from 5x minus 5 i can take out a 5 and i have x minus 1 left and then as long as you did everything right this will always work out where you have a common factor here that you can take out now so we have x minus 1 times two x plus five which is what i got when i did guess and check okay it's strategic guess and check though if you need a review on how to do this whole ac method and splitting the middle term i'll link a video here all on factoring okay from this point now we do the zero product property so x minus one is equal to zero or two x plus five is equal to zero that means x is one or two 2x is negative 5 which means x is negative 5 halves so were either of those restrictions let's see negative 4 and 2 nope those are not restrictions so then i'm going to go ahead put them in my solution set my solutions are 1 and negative 5 halves okay now i told you we were going to do two methods the other method involves the quadratic formula and it's very useful because not every quadratic equation factors so you're going to just have to memorize it if you haven't already just like with factoring you need to make sure that you set the quadratic equation equal to zero first before you use the quadratic formula okay so x equals when i read it out loud i say the opposite of b plus or minus the square root of b squared minus 4ac and then be careful that you divide the whole thing by 2a all right so one little warm-up i threw in an extra one before we look at kind of the spicier college algebra level ones coming up so solve this equation we're going to use the quadratic formula before i get rolling though i got to set it equal to zero so let's subtract 13. so x squared minus 6x minus 13 equals zero let's list out what a b and c are okay so a is the coefficient on x squared it's just a one b is the coefficient of x it's negative 6 and c is the constant negative 13. okay always put what your variable is in this case x equals opposite of b so negative negative six plus or minus the square root of b squared so negative six squared minus four times a which is one times c which is negative thirteen all over two a two times one okay then we just clean up so this equals positive six plus or minus square root negative 6 squared is going to become positive 36 and then watch this negative 4 times 1 times negative 13 is going to give me a positive number right so 4 times 13 that's 52 and then this whole thing gets divided by 2. now the next step you're always going to clean up underneath the radical so this is 6 plus or minus square root 36 plus 52 that's 88 okay over 2. we got to make sure we simplify our radical so 88 that's 4 times 22 so that's 6 plus or minus square root of 4 times 22 over 2. so this is going to be 6 plus or minus remember you can break up that radical round four grad twenty-two so two rad twenty-two over two and then we can cancel out a two from everybody please do so carefully i'll show you two options you can split the denominator so you could write this as 6 over 2 plus or minus 2 rad 22 over 2 and then you end up with 3 plus or minus this is gone rad 22 that's one way other way factor out a common factor from the numerator the greatest common factor i can take out a 2 and then i'm left with 3 plus or minus rad 22 over 2 cancel cancel and then this is the same thing that we got here all right what i don't want you doing is right here just slashing out the twos like this because you'll probably forget that you also have to cancel out a two from the six like 99 of students who just do this little quick method forget about this guy okay so either split the denominator like this or factor this is probably the better way okay to do it so that's it there were no restrictions these are our solutions and notice we couldn't have um factored because we have an irrational solution that radical 22. so we had to use the quadratic formula factoring is typically faster and easier whenever your equation factors but quadratic formula will work for ones that factor as well okay so moving on now let's look at a couple more tricky ones first thing here example three a we have two thirds x squared minus x minus three equals zero now there's no restrictions i know i i have a denominator i know but it's just a constant there's no variables in that denominator so we don't have to stress about restrictions so i'm just going to get this thing fraction free i'm going to multiply everybody by 3 the lcd before i use the quadratic formula yes so 2 3 x squared times 3 is just going to be 2x squared and then we're going to have minus 3x minus 9 and 0 times 3 is still 0. so let's list out what a b and c are before we get rolling so a is 2 b is negative 3 and then c is negative 9. all right so here we go x equals opposite of b so that would be positive 3 plus or minus the square root of b squared so negative 3 squared minus 4 times a times c so four times two times negative nine all over two times two nice so this is going to be three plus or minus square root that's nine and then since i have two negatives here being multiplied it's gonna turn into positive four times two that's eight times nine is seventy-two over 4. so this is going to be 3 plus or minus square root 9 plus 72 that's 81 over 4. well square root of 81 is 9 isn't it 3 plus or minus 9 over 4. do you leave it like this no because we don't have a radical stuck in the equation 3 plus 9 is 12 over 4 or 3 minus 9 is negative 6 over four and those reduce so we get either twelve over four that's three negative six over four is negative three halves we had no restrictions so those go in the solution set so notice here since our answers were rational numbers we didn't have something like rad 23 left over like last time we actually could have solved this one by factoring but we're just practicing the quadratic formula right now so just pay attention when you're working on your exercises or you're doing um problems in class or on a test to use the method that the instructor says if they let you pick then you pick okay b 2x over x minus three plus one over x equals four so we are gonna have restrictions here x cannot equal three or zero are you guys getting more comfortable finding the restrictions and then i do want to clear out the denominator so i want to multiply by the lcd the lcd is going to be x minus 3 times x so let's rewrite everything 2x over x minus 3 plus 1 over x equals 4 then i'm going to multiply by the lcd so x minus 3 x x minus 3 x and then one more here x minus 3 x okay cancel cancel i just have 2x times x so that's 2x squared plus let's look at the next term x's cancel so i just have 1 times x minus 3 equals i'm going to write this as 4 x times x minus 3. you guys cool with that okay good so let's clean up and then set equal to zero before we start figuring out a b and c this is gonna be four x squared minus twelve x and typically you don't want a to be negative when using the quadratic formula it just makes life so much more complicated so i'm going to um have a 0 on the left-hand side so i'm going to subtract 2x squared subtract x plus 3. subtract two x squared plus x plus three so now we've got zero equals two x squared let's see here oh i should have subtracted x forgive me minus x so that'll be minus 13x plus 3. okay now we can identify a b and c so a is 2 b is negative 13 c is 3. why don't you pause the video see if you can do this next part on your own plugging in everything and simplifying okay did you try i hope you did so x is equal to opposite of b plus or minus square root b squared negative 13 squared is going to be positive 169 yeah minus 4 times a times c all over 2a 2 times 2. so we're going to have x equals 13 plus or minus square root that's 169 minus 24 so 145 over 4. and actually radical 145 doesn't simplify so we're done so let me just erase this and then put a nice little solution set on it and neither of those were our restrictions our restrictions were 3 and zero so we're in the clear okay good so one last topic that we're going to study now is the discriminant and the discriminant of a quadratic equation is the expression b squared minus 4ac and your book uses this symbol here which sometimes you know refers to change or delta so i don't know why they're using it for the discriminant but whatever maybe because they got tired of writing discriminant um it's basically the stuff or the expression underneath the square root sign in the quadratic formula so let's look back at the quadratic formula that we had in our notes this right here is the discriminant just that part underneath without the square root so if they ask you to find the discriminant you're just finding this part right here that i'm circling okay so here's some things to keep in mind about the discriminant it'll tell you what kind of solutions you're going to have so basically if it's greater than 0 if it's positive then your equation has two different real solutions like here 3 negative 3 halves what was the discriminant it was 81. we love it if the discriminant is 0 the quadratic equation has a real solution of multiplicity 2. i'll show you what that is going to look like because it can be a little confusing and then third is if the discriminant is less than zero so if it's negative then the quadratic equation has two different complex solutions meaning it has no real solutions we're going to work on figuring out what those are later but basically that's because if you have a negative quantity underneath the radical if this stuff's negative then you won't have a real solution you'll have a solution in the complex field of numbers so it'll involve the imaginary unit i which we're going to study coming up okay all right so don't stress the homework exercises or the problems that you do with the discriminant aren't too bad look what the directions say use the discriminant to determine whether each equation has two real one real or two complex solutions without solving the equation so you're not solving you're just finding the discriminant and then saying are you in case one two or three that's it okay so here we can use delta since they're so keen on it um a is one b is 4 and c is 7. right everything's set equal to 0. so delta or the discriminant equals b squared minus 4 times 1 times 7. so this is 16 minus 4 a c so that's going to be 4 times 1 times 7 that's just 28 okay so 16 minus 28 i mean i don't really care what it is i already can tell it's negative it's less than zero so i'm in this scenario case three don't put case three though i mean that's just what your book calls it so you have to say that we have two different complex solutions or no real solution okay good let's look at the next one example b a is equal to two b is equal to negative three and c is negative seven so discriminant is gonna equal b squared minus 4 a c so this is going to be positive 9 and then since we have two negatives here this is going to be plus 4 times 2 is 8 times 7 56. so 9 plus 56 that's going to be 65 that's greater than zero so what does that tell me i'm in case one where we have two different real solutions okay one more one more c let's do x squared minus 10 x plus 25 equals zero so let's list out what a b and c are a is 1 right b is negative 10 and then c is 25. so discriminant b squared negative 10 squared are you are you putting the parentheses nicely like i am watch pay attention you have to put the parentheses around the negative 10 and the exponent goes outside that way you know that it's going to become positive right after you square it minus 4 times a times c so negative 10 squared that's a hundred minus four times one times 25 y that's a hundred as well so this is zero when the discriminant is zero remember that's that second case where you have a real solution multiplicity two and i wanna show you what that means okay so the equation was x squared minus 10 x plus 25 equals zero if if you were to try to solve let's say we're going to solve by factoring not the quadratic formula this cute little guy factors it's going to factor into x minus five times x minus five right good so far okay and then you do zero product property x minus five is zero but wait a minute this is the exact same thing so you get x equals 5 twice that's what it means that it's multiplicity 2. and you don't write it twice you know that's pointless so usually you just put it in the solution set by itself once but we know it's a quadratic equation it's going to have two solutions so since it's in there twice it counts as double basically and another cool way to kind of think about it is you don't have to factor writing x plus x minus 5 twice like that right couldn't we have written it as x minus 5 quantity squared and that little exponent that tells me the multiplicity it's 2. so that solution is in there twice that's what it means and that happens with the quadratic equation any time the discriminant is zero that's how you know you have that special case where you got that double root or double solution okay so i wanted you to see one of each and then don't worry later on you're going to look how to solve the ones that have complex solutions so you can feel really fulfilled and proficient so give the video a thumbs up if you found it helpful stay tuned we're going to look at solving more equations pretty soon