In this video, we're going to focus on factoring trinomials the easy way. So, let's start with this one. x squared plus 5x plus 6. How can we factor it? So notice that the leading coefficient is 1. When you see that, look at this number 6. Find two numbers that multiply to 6 but add to the middle term 5. So let's make a list of the numbers that multiply to 6. 1 times 6 is 6. 2 times 3 is 6. But only 2 times 3, I mean 2 plus 3 adds to 5, so we need to use 2 and 3. So the answer is going to be x plus 2 times x plus 3. That's not bad, is it? Now, if you FOIL it, you can check your answer.
x times x is x squared. x times 3, that's 3x. 2 times x is 2x, and 2 times 3 is 6, so if you add the two middle numbers, 3x plus 2x, that's going to give you 5x. So we can get our original expression, so we know that this is the correct answer.
So that's a quick and simple way in which you can factor a trinomial when the leading coefficient is 1. But now let's go over another example. Let's say if we have x squared plus 8x plus 15. So feel free to pause the video and see if you can factor this expression, and then unpause it and check your answer. So what two numbers multiply to 15 but add to the middle number 8?
So we have 1 and 15, and 3 and 5. Now 1 plus 15 is 16, so that's not going to work. But 3 plus 5 does add to 8, so that can work. So the answer is going to be x plus 3. times x plus 5. Let's try another one. Let's try x squared minus 7x plus 12. So what two numbers multiply to 12 but add to negative 7? So let's make a list.
12 divided by 1. is 12. 12 divided by 2 is 6. 12 divided by 3 is 4. Now, 3 plus 4 adds to positive 7, but negative 3 and negative 4 adds to negative 7. But negative 3 times negative 4 is positive 12. And so the answer is going to be x minus 3 times x minus 4. Now, what about this one? x squared plus 3x minus 40. How can we factor this expression? So what two numbers multiply to negative 40 but add to 3? So negative 40 divided by 1 is negative 40. If we divide it by 2, we'll get negative 20. 3 doesn't go into it. If we divide it by 4, we'll get negative 10. And if we divide it by 5, we'll get negative 8. Now, 5 plus negative 8 is negative 3. So if we change the sign, it can work.
Negative 5 plus 8, that's positive 3. So the answer for this problem is x minus 5 times x plus 8. And that's equal to x squared plus 3x minus 40. So now, what will you do if the leading coefficient is not 1? Let's say if we want to factor 2x squared minus 3x minus 2. So how can we factor this expression? And let's say if it's equal to 0, how can we solve for x? So the first thing you need to do is multiply the first and the last term.
So 2 times negative 2, well that's equal to negative 4. So we need to find two numbers that multiply to negative 4 but add to the middle term negative 3. So this is going to be negative 4 and 1. Negative 4 times 1 is negative 4, and negative 4 plus 1 is negative 3. So what we're going to do is we're going to replace the middle term, negative 3x, with negative 4x plus 1x. And then after that... We're going to factor by grouping. So in the first two terms, let's take out the GCF, the greatest common factor.
In this case, it's going to be 2x. To find out what goes inside, divide 2x squared by 2x. 2x squared divided by 2x is x, and negative 4x divided by 2x, that's a negative 2. Now, there's no GCF between 1x and negative 2, so in such a situation, factor out a 1. So, we're just going to get x minus 2. Now, if these two are the same, that means you're going in the right direction.
So let's factor x minus 2 from both terms. So if we take an x minus 2 from this term, what we have left over is 2x. And if we take it here, what we have left over is plus 1. So this is the answer, x minus 2 times 2x plus 1. Now granted, it's still equal to 0, so we can solve for x. But now you know how to factor a trinomial whenever the leading coefficient is not 1. So at this point, if you want to... solve for x set each factor equal to 0 so if the one on the left side we can just add 2 to both sides and we can get the answer x is equal to 2 so that's one solution to get the other solution we need to subtract track one from both sides initially and we'll have 2x is equal to negative 1. To separate the two from the x we need to divide both sides by 2 so x is equal to negative 1 half so that's the second solution.
Now what about this one? 3x squared plus 8x minus 3. Go ahead and factor the expression, solve for x, and then once you finish, unpause the video to see if you have the right answer. first thing we're going to do we're going to multiply the 3 times the negative 3 and that's going to give us negative 9 now we need to find two numbers that multiply to negative 9 but add to 8 since negative 9 differs from 8 by 1 We need 1 as one of our answers.
So 1 and negative 9, that's going to add to negative 8. But negative 1 and 9, that's going to add to positive 8. So those are the numbers. So let's replace 8x with 9x and negative 1x. So let's factor by grouping.
If we take out the greatest common factor between 3x squared and 9x, that's going to be 3x. 3x squared. divided by 3x is x and 9x divided by 3x that's 3 and for the last two all we can take out is simply a negative 1 so negative 1x divided by negative 1 that's positive x and negative 3 divided by negative 1 is positive 3 so we have the common factor x plus 3 now the 3x on the outside is going to go in here and the negative 1 is also going to go in here as well And so that's how you can factor it.
Now it's still equal to 0, so we need to solve for x at this point. So let's set each factor equal to 0. So for the one on the left, let's subtract 3 from both sides. And so these will cancel.
And one of our answers is... x is equal to negative 3 and for the next one let's add 1 to both sides and so what we have now is 3x is equal to 1 so now we need to divide both sides by 3 So x is equal to 1 third. So those are the two answers that we have for that particular equation. Now what about this one?
Let's say if we have 4x squared minus 4x minus 3. And let's say that's equal to 0. Go ahead and solve for x by factoring. So let's multiply 4 and negative 3. That's equal to negative 12. And let's find two numbers that multiply to negative 12 but that add to negative 4. So, let's make a list. We have 1 and 12, 2 and 6, 3 and 4. Now 2 plus negative 6, that adds to negative 4. So let's replace negative 4x with positive 2x and negative 6x.
And now let's factor by grouping. So let's take out 2x from the first two terms. 4x squared divided by 2x.
well that's 2x, and 2x divided by 2x is 1. Now for the last two terms, the greatest common factor that we can pull out is negative 3. Negative 6x divided by negative 3 is 2x, and negative 3 divided by negative 3, that's plus 1. So we have a common factor 2x plus 1. And so now we need to insert the 2x and the negative 3 in the other term. other parentheses. So now, we could set 2x plus 1 equal to 0, and 2x minus 3 equal to 0. So 2x is equal to negative 1, if we divide by 2, x is negative 1 half, that's the first answer.
If we add 3 to both sides, we'll get this, divide by 2, x is 3 halves. So now sometimes, if you ever have difficult... factor and expression you can always use the quadratic equation so let's use the quadratic equation for this example x is equal to negative B plus or minus B squared minus 4ac divided by 2a So the number in front of x squared is a, in front of x is b, and this is c. So we have negative, and b is negative 4, plus or minus b squared, negative 4 times negative 4, that's 16, minus 4 times a, which is 4, and c is negative 3, divided by 2a, or 2 times 4. So negative 4 times negative 4, that's 4. And let's see, 4 times 4 is 16, 16 times 3 is 16. three that's going to be 48 but because we have two negative signs it's going to be positive 48 and on the bottom to divide two times four is eight so let's make some space always run out of space so 16 plus 48 that's 64 and 64 the square root of 64 is 8 so now we have two answers have 4 plus 8 divided by 4 and 4 minus 8 divided by 4. 4 plus 8 is 12 and 12 over 4 is equal to, well actually, I don't know why my 4 became an 8. So let's fix that.
This should have been an 8. Okay, so we're going to have two equations, 4 plus 8 divided by 8 and 4 minus 8 divided by 8. So 4 plus 8 is 12. 12 and 4 minus 8 that's negative 4 so we've got to reduce the fraction by dividing by 4 12 divided by 4 is 3 8 divided by 4 is 2 so that's one answer that we had before which was 3 over 2 and now for negative 4 over 8 let's divide the top and bottom by negative 4 negative 4 divided by 4 is negative 1 and 8 divided by 4 is 2 so that's the other answer that we had before and so you can get the same answer if you use the quadratic equation.