>> Welcome to the Cypress College math review on solving quadratic equations by factoring.
In objective one, you will learn how
to solve quadratic equations having two terms using factoring.
Our strategy has four steps.
First, gather all terms on one side of the equation if needed.
Second, factor, third, apply the zero-product property and fourth, solve for the variable.
So, let's get started.
In this example, we have to solve 27y squared equals 36y.
Remember our strategy, step one says to gather all terms on one side of the equation if needed.
We definitely need to do that in this equation because we have terms on both sides.
So, let's bring the 36y over to the left-hand side
and we would have 27y squared minus 36y equals zero.
Now it's important to remember when you're solving equations
that every step should have a left-hand side, an equal sign, and a right-hand side.
Let's move on to step two which says we should factor.
Now between these two terms, we see that we have a greatest common factor of 9y
and we would have left 3y minus 4 and then of course our equal sign, and we always have
to have a right-hand side which, in this case, is zero.
Now let's double check our factoring, 9y times 3y gives us 27y squared
and 9y times a negative four gives us negative 36y.
Step three says to apply the zero-product property.
Now the zero-product property says that, if A times B is equal to zero,
then A is equal to zero, or B is equal to zero.
The only way that two numbers multiplied together can equal zero is
if one of the numbers is zero.
Now in our case the 9y is acting like the A and the 3y minus 4 is acting like the B.
If these two factors multiplied together equals zero,
then that means one of them has to equal zero.
So, when we apply the zero-product property, we get that 9y would have to equal zero,
or 3y minus 4 would have to equal zero.
Step four now says to solve for the variable.
So, in this first equation, if we divide both sides by 9, we would end up with y is equal
to zero so that's our first solution.
In the second equation, we would add four to both sides and get 3y is equal to four.
Then we would divide both sides by 3 and get that y is equal to 4/3.
And that is our second solution.
I want to share with you now a tip about solving equations and help you avoid this common error.
Never divide both sides by the variable.
You always have to gather all terms to one side as we did here in the first step.
Now here's why you can't do that.
Suppose we started off with our equation and divided both sides by y. We would end
up with 27y equals 36, then you would divide both sides by 27,
you have 36/27 which reduces to 4/3.
Now notice here, that while we got one of the solutions which is why equals 4/3,
we are completely missing the second solution to this equation which is y equals zero.
So, when you make the error of dividing by the variable in the first step,
that leads to losing one of the solutions to the equation.
Another way to think about it is, you know you can't divide by zero ever.
And look at what one of the solutions to this equation is, zero.
So, if you were to divide by y in that first step, there you are trying to divide
by zero which you know is invalid.
So, always gather all the terms onto one side of the equation and then factor
and use the zero-product property.
In this example, we have to solve 16a squared minus 25 is equal to zero.
Now, in our strategy, step one is to gather all terms on one side of the equation if needed.
Well that's already done for us, so let's move on to step two where we have to factor.
We recognize that the left-hand side is the difference of squares
so we can factor immediately into the quantity 4a plus 5 times the quantity 4a minus 5.
We bring down the equal sign, we bring down the zero.
Step three tells us to apply the zero-product property.
Now we know that means we take each of the factors and set them equal to zero.
So, we have 4a plus 5 is equal to zero or 4a minus 5 is equal to zero.
Step four says to solve for the variable.
In this first one, we would subtract 5 from both sides and have 4a is equal to negative 5,
and then we would divide both sides by 4 and end up with A is equal to negative 5/4.
In the second equation, we would add 5 to both sides, so we have 4a is equal to 5.
Then we divide both sides by 4 and we end up with A is equal to 5/4.
And these are the two solutions to this equation.
It's time to check your understanding.
So pause the video to try this one on your own,
then restart when you are ready to check your answer.
In this example, we have to solve 12h squared is equal to 21h.
Let's refer to our strategy.
Step one is to gather all terms on one side of the equation if needed,
and we definitely need to do that in this one.
So, we will bring everything to the left-hand side
and have 12h squared minus 21h is equal to zero.
Step two says to factor.
Now, here we see that the greatest common factor is 3 times H,
and what we would have left is 4h minus 7.
Remember to bring down your equal sign and bring down the zero.
Just a quick reminder that every step of every equation has to have a left-hand side,
an equal sign, and a right-hand side.
Now let's double check our factoring, 3h times 4h is 12h squared,
and 3h times a negative 7 is negative 21h.
Step three says to apply the zero-product property.
So, we are going to take each of these factors and set it equal to zero.
Then we would have 3h is equal to zero or 4h minus 7 is equal to zero.
We move on to step four which says to solve for the variable.
In this first equation, we will divide both sides by 3 and end up with H is equal
to zero and that's our first solution.
In the second equation, we would add 7 to both sides so that now we have 4h is equal to 7,
and then we would divide both sides by 4 and end up with H is equal
to 7/4, and that is our second solution.
Another tip about solving equations is that, whenever you get your answers,
you can always know whether you did the problem right or not,
because you can always check your answers.
So, for example, let's do that here.
If we checked H is equal to zero, what we would do is plug that in to the original equation
and double check that 12 times zero squared does equal 21 times zero,
and because the left-hand side is zero
and the right-hand side is zero, that means this one did work.
We could also check the H equals 7/4 answer in the original.
So, we would have to show that 12 times 7/4 squared is equal to 21 times 7/4.
Well, on the left-hand side, we have 12 times 49/16.
On the right-hand side, we know that 21 times 7 would be 147/4.
On the left-hand side, the 12 and 16 have a common factor of 4, so let's reduce that.
Four goes into 12 three times and 4 goes into 16 four times and 3 times 49 is 147.
The denominator is 4 so, those two worked out.
So, in this way, you can always check your answers
and know whether you did the problem correctly or not.
Below are some practice problems you can try on your own.
You may either pause the video to work on them now or write them down to work on them later.
After a few seconds, the answers will be revealed.
In objective two, you will learn how to solve quadratic equations using foil factoring.
Let's look at our strategy.
Step one will be to gather all terms on one side of the equation if needed.
Step two will be to factor using either trial and error or the AC method.
Step three, apply the zero-product property, and step four, solve for the variable.
In this example, we have to solve 2x squared equals 7x plus 15.
If we consult our strategy, step one is to gather all terms
on one side of the equation if needed.
You can see we have terms on both sides of the equation.
Now, because the coefficient of the x squared is positive on the left-hand side,
I'm going to bring the right-hand side terms over to the left-hand side to keep
that leading coefficient positive.
Then this will become 2x squared minus 7x minus 15 is equal to zero.
Step two says to factor and we will be using trial and error method in this example.
So, the factors of 2 are 1 times 2 and the factors of 15 are 1 times 15 or 3 times 5.
So, the only way we're going to get this 2x squared is to take X times 2x.
Then we have some choices for the constant term,
typically we will choose the factors that are closer together.
So, we'll say 3 and 5.
Now we know one of them will be negative and one will be positive
because the constant term is negative.
But we will wait until we do our cross-product terms to pick which one.
The first cross-product is 6x and the second cross-product is 5x.
Now because the middle term is negative,
we will always pick the larger magnitude cross-product to be the one that's negative.
So, I would put the negative here which means the 3 is negative.
Now, when we add up these cross-products, we get negative X which does not equal the negative 7x
that we need, so that tells us we have to try something else.
So, if we keep the X and the 2x going, maybe we will try switching the 3 and the 5.
So, we'll put the 5 up here and the 3 down here.
The first cross-product is 10x, the second cross-product is 3x and, remember,
we said we'll pick the larger magnitude cross-product and make it negative.
So that's the 10x which means the 5 would have to be negative.
Now when we add these two together, we get negative 7x
which is exactly what we needed for the middle term.
Now, that tells us that our factors are going
to be the quantity X minus 5 times the quantity 2x plus 3.
And, of course, that still is equal to zero.
Step three says to apply the zero-product property, so we will take each
of these factors and set them equal to zero.
So, X minus 5 could be zero or 2x plus 3 could be zero.
Step four is to solve for the variable.
In this first equation, we add 5 to both sides and get that X is equal
to 5, so that our first solution.
In the second equation, we would subtract 3 from both sides and get 2x is equal to negative 3.
Then we would divide both sides by 2 and end up with X is equal to negative three halves.
And that is our second solution.
And, if you have time, remember it's always a good idea to check your answers
by plugging them back into the original equation.
For the purposes of this lesson though, we are going to move on.
In this example, we have to solve 6n squared minus 25n is equal to negative 4.
From our strategy, we know that step one is to gather all terms
on one side of the equation if needed.
So, we will choose to move this negative 4 over to the left-hand side by adding 4 to both sides.
So, we'll end up with 6n squared minus 25n plus 4 is equal to zero.
Step two says to factor and, in this case, we'll be using the AC method.
To do that, we make the little cross grid here, we know that the AC term goes
in this upper area meaning the leading coefficient of 6 has
to multiply the constant term of 4, and we get a positive 24.
Whatever the B term is, the middle term, that coefficient which is negative 25 goes down here.
So, we need two numbers that multiply to be a positive 24, but add to be a negative 25.
And those two numbers would be negative 1 and negative 24.
Now, these two numbers tell us how to break up the middle term.
So, it will become negative N minus 24n, the 6n squared comes straight down.
And the constant term of 4 comes straight down, equals zero.
Now we factor 2 by 2.
From these first two terms, we can factor out an N and we would have left 6n minus 1.
Let's check it.
N times 6n is 6n squared, N times a negative 1 is negative N. Whatever this math symbol is,
you bring it straight down, so that's a minus there.
From these two terms, we can factor out a 4.
Now, you want to be careful because, remember, it's really negative 4
so then what we would have left is 6n minus 1.
Let's check our factoring, negative 4 times 6n is negative 24n,
negative 4 times negative 1 is positive 4.
We have to bring down the equal sign and bring down the right-hand side which is zero.
Now from these two terms, we can factor out 6n minus 1
and what we have left is N minus 4 equals zero.
Step three says to apply the zero-product property,
so we will do that in this next step here, where we take each
of these factors and set it to zero.
So, we have 6n minus 1 is equal to zero or N minus 4 is equal to zero.
Step four says to solve for the variable.
In this first equation, we would add one to both sides and get 6n is equal to 1,
and then we would divide by 6 and get that N is equal to 1/6.
That's our first solution.
In the second equation, we would add 4 to both sides and get N is equal
to 4, and that is our second solution.
It's time to check your understanding, so pause the video to try this one on your own,
then restart when you are ready to check your answer.
In this example, we have to solve 2 is equal to 10k squared plus K. In our strategy,
step one is to gather all terms on one side of the equation if needed.
Now the only thing we need to move here is this 2, so we will subtract 2 from both sides
of the equation and get zero is equal to 10k squared plus K minus 2.
Step two is to factor using trial and error or the AC method.
And I think we'll go ahead and use trial and error in this example.
When we look at the factors of 10, we know it could be 1 times 10 or 2 times 5 and,
for the factors of 2, it's going to be 1 times 2.
Now in deciding which of these to try first, we typically will start
with the factors that are closer together.
So, let's do 2k and 5k.
Now the only way to get 2 is with a 1 times a 2 because that 2 is negative,
we know that one of these will be negative and one will be positive, but we will wait to decide
until after we do our cross-products.
The first cross-product is 5k, the second cross-product is 4k.
Now, because the middle term is positive,
we are going to want the larger magnitude cross-product to be positive.
So, that means we'll have the 5k be positive, we'll want to make the 4k negative
which means that 2 has to be negative.
Now, when we add these up, we get k
which already is exactly what we needed for our middle term.
So, then we've got zero is equal to,
now our factors will be the quantity 2k plus 1 times the quantity 5k minus 2.
Step three is to apply the zero-product property which tells us that we have
to set each of these factors equal to zero.
So, we would have 2k plus 1 has to equal zero or 5k minus 2 has to equal zero.
Step four is to solve for the variable.
In this first equation, we would subtract 1 from both sides and get 2k is equal to negative 1,
and then we would divide by 2 so we'd get K is equal to negative ½. In the second equation,
we would add 2 to both sides, so we have 5k is equal to 2 and then we would divide
by 5 to get that K is equal to 2/5.
And these are our two solutions.
Below, are some practice problems you can try on your own.
You may either pause the video to work on them now, or write them down to work on them later.
After a few seconds, the answers will be revealed.
In objective three, you will learn how
to solve quadratic equations with the greatest common factor.
Our strategy has five steps.
Step one is to gather all terms on one side of the equation if needed.
Step two, factor out the GCF, the greatest common factor.
Step three, factor the quadratic using trial and error or AC method.
Step four, apply the zero-product property, and step five, solve for the variable.
So, let's get started.
In this example, we have to solve 30x minus 16x squared is equal to 24x cubed.
In our strategy, step one is to gather all terms on one side of the equation if needed.
Because we have terms on both sides, we will definitely need to do that,
and I'm going to choose to take the left-hand side terms and move them over to the right,
because my leading coefficient of 24 is already positive there.
So, the left-hand side will become zero, the right-hand side we keep that 24x cubed.
Now, over here, we are subtracting 16x squared, so that means we'll have
to add 16x squared to both sides.
And then, here, we will have to subtract 30x from both sides.
Step two says to factor out the GCF, which we know is the greatest common factor.
So, we bring down the zero on the left-hand side, bring down the equal sign and,
among these three terms, our GCF is going to be 2x
and we would have left 12x squared plus 8x minus 15.
Always double check that 2x times 12x squared is 24x cubed, 2x times 8x is 16x squared,
and 2x times a negative 15 is negative 30x.
Now step three says to factor the quadratic using trial and error or AC method, and I think,
for this example, we'll go ahead and use trial and error.
Let's take a look at the leading coefficient of 12, and we know it could be 1 times 12,
it could be 2 times 6, or 3 times 4.
For the constant term of 15, it could be 1 times 15, or 3 times 5.
Now, typically, we will try starting out with the factors that are closest together.
So, maybe for the X squared term, we'll do 3x and 4x,
and for the constant term, we'll do 3 and 5.
Now because the 15 is negative, we know that one of these will be positive
and one will be negative, but let's do the cross-products first.
The first cross-product is 12x, the second cross-product is 15x.
Because this middle term is positive, we know that means we want
to make the larger magnitude cross-product positive,
which means make the smaller one negative.
If this is negative, then that means we have to make that 3 negative.
Now when we add these two, we get 3x but, unfortunately,
that does not equal the 8x that we need.
So, maybe we'll try switching the 3 and the 5 so, 3x and 4x will stay the same.
And then we'll put the 5 here and the 3 here.
The first cross-product would be 20x, the second cross-product would be 9x.
We know we're keeping the bigger one positive and taking the smaller one, making it negative.
Now, when we add these, we get 11x, but that does not equal the 8x that we need.
So, maybe we need to try switching these first ones.
So, maybe we'll try 2 and 6.
We've got 2x and 6x and we'll try putting the 3 and the 5.
Now, the first cross-product is 18x, the second cross-product is 10x.
We keep the larger one positive, make the smaller one negative,
which means we have to make that 5 negative.
Now when we add these two up, we get 8x
which does match exactly what we need in the middle term.
So, this tells us that our equation will become zero equals,
take this 2x and bring it straight down.
And now, this quadratic is going to factor
into the quantity 2x plus 3 times the quantity 6x minus 5.
Step four is to apply the zero-product property.
Now this time, we have three factors being multiplied together and getting zero.
So, that means, when we apply the zero-product property,
we have to take all three and set them to zero.
So, we would have 2x is equal to zero, or 2x plus 3 is equal
to zero, or 6x minus 5 is equal to zero.
Step five says to solve for the variable.
So, let's take these one at a time.
In this first equation, we would divide by 2 and get that X is equal to zero.
In the second equation, we would subtract 3 so now we have 2x is equal to negative 3.
And then we would divide both sides by 2 and get that X is equal to negative three halves.
In this third equation, we would add 5 to both sides, so we have 6x is equal to 5,
then we would divide both sides by 6 and we would get that X is equal to 5/6.
And these are the three solutions to this equation.
In this example, we have to solve 120y cubed minus 44y squared minus 8y is equal to zero.
Referring to our strategy, step one says to gather all terms
on one side of the equation if needed.
Well, that's already been done for us, so we can move on to step two which says to factor
out the GCF, the greatest common factor.
Now for these three terms, we can see that we would have a greatest common factor of 4y.
And what we would have left is 30y squared minus 11y minus 2, close the bracket,
bring down the equal sign, and the zero.
Step three says to factor the quadratic using trial and error or AC method.
Here's our quadratic.
I think we'll use the AC method in this one, so we practice it as well.
So, we make our little X grid, we know
that the AC term would be 30 times a negative 2, so that would be negative 60.
And the B term is negative 11, so we need two numbers that will multiply
to be negative 60, but add to be negative 11.
Now the best way to not miss any possibilities is to just go through the factors in order.
So, if we try 1 and negative 60, that adds to negative 59 instead of negative 11.
Two and negative 30 adds to negative 28, again not what we need.
Three and negative 20 adds to negative 17, and we can see that 4 and negative 15 would multiply
to be negative 60, but add to be negative 11.
Remember that these two terms tell us how to break up the B term into plus 4y and minus 15y.
The 30y squared comes straight down.
And the minus 2 comes straight down.
We also have to bring down that GCF of 4y,
and then we have to put our equal sign and the right-hand side of zero.
Then we would bring down the 4y and we would look to factor these two by two.
So, from these first two, we see that we can factor out a 2y
and we would have left 15y plus 2.
Whatever math symbol is here, you bring it directly down, so that's a minus sign.
Now from these two terms, there's nothing that can be factored out other than 1 and, of course,
it's a minus 1, so then that means we would have left 15y plus 2, close the bracket, equals zero.
So, let's double check this whole step.
2y times 15y is 30y squared, 2y times 2 is plus 4y negative 1 times 15y is negative 15y
and negative 1 times 2 is negative 2.
Now from these two terms, we see that we can factor out 15y plus 2
and we would have left the quantity 2y minus 1.
We have to bring down that GCF of 4y.
We have to bring down our equal sign and the zero.
Step four says to apply the zero-product property.
We have three factors, and we have to take each of them and set it equal to zero.
So, we will have 4y is equal to zero, or 15y plus 2 is equal
to zero, or 2y minus 1 is equal to zero.
Step five says to solve for the variable.
In this first equation, we would divide both sides by 4
and end up with Y equals zero, there's our first solution.
In the second equation, we would subtract 2 from both sides,
so now we have 15y equals negative 2.
Then we would divide both sides by 15 and end up with Y is equal to negative 2/15.
And, in this third equation, we would add 1 to both sides, so that we have 2y equals 1,
then we would divide by 2 and end up with Y is equal to ½.
And these are the three solutions to this equation.
It's time to check your understanding, so pause the video to try this one on your own,
then restart when you are ready to check your answer.
In this example, we have to solve negative 8p cubed plus 18p squared is equal to 7p.
In our strategy, step one is to gather all terms on one side of the equation if needed.
We definitely need to do that here and, because this leading coefficient is negative
on the left-hand side, I'm going to choose to move all the terms
from the left-hand side over to the right-hand side.
So, then that means this left-hand side will become zero and the right-hand side we'll have
to add 8p cubed to both sides so it becomes positive 8p cubed.
We would subtract 18p squared and we bring down the positive 7p.
Step two says to factor out the GCF, so bring down the zero and the equal sign
from these three terms, we see that our GCF is P
and what we would have left is 8p squared minus 18p plus 7.
Step three says to factor the quadratic using trial and error or AC method.
Let's use trial and error for factoring this quadratic.
For the factors of 8, we could do 1 times 8 or 2 times 4.
And for the factors of 7, there's only 1 times 7.
Typically, in trial and error, we choose to start
with the two factors that are closer together.
So we will try 2p times 4p.
And then, for the constant term, we know it's going to be 1 and 7.
Now, because the constant term is positive but the middle term is negative,
we know that both the 1 and the 7 will have to be negative.
So, let's check our cross-products now.
The first cross-product is negative 4p, the second cross-product is negative 14p and,
when we add those up, we get negative 18p
which already is exactly what we need for the middle term.
So, let's bring our equation down, zero equals, we have to bring that GCF of P down.
And now what we discovered was this quadratic is going to factor
into the quantity 2p minus 1 times the quantity 4p minus 7.
Now step four tells us to apply the zero-product property.
And we know that that means we have to take each of these three factors and set it to zero,
so we have P equals zero or 2p minus 1 equals zero, or 4p minus 7 equals zero.
Step five says to solve for the variable.
In this first equation, we're already done solving, so that solution is P equals zero.
In the second equation, we would add 1 to both sides, so now we have 2p equals 1.
Then we would divide by 2 and get P is equal to ½. In the third equation,
we would add 7 to both sides and have 4p is equal to 7.
Then we would divide by 4 and have that P is equal to 7/4.
And these are the three solutions to this equation.
Below, are some practice problems you can try on your own.
You may either pause the video to work on them now or write them down to work on them later.
After a few seconds, the answers will be revealed.