Video: Factorizing a Trinomial Using Guess-and-Check

Chris O’Reilly

Factor 𝑥² + 8𝑥 − 20.


Video Transcript

Factor 𝑥 squared plus eight 𝑥 minus 20.

So when we’re looking at this question, what we need think about is what does factor mean. Well, factor actually means putting our expression into parentheses. And because our expression has an 𝑥 squared term, an 𝑥 term, a numerical term, and then this means that actually what we’re gonna have is two pairs of parentheses because we’re actually factoring a quadratic.

Okay, so now, we actually have our two pairs of parentheses are put in here. Let’s start thinking about the terms that are gonna go inside. Well, first of all, we know that we have an 𝑥 at the beginning of each parenthesis. That’s because we need to actually make 𝑥 squared and its positive 𝑥 squared. So what we’re gonna have is 𝑥 multiplied by 𝑥 to give us the 𝑥 squared.

Okay, now, let’s move on to what else is gonna be inside our parentheses. Okay, so when we’re actually factoring a quadratic, what we look at first is we go right. We need to find two factors, where the sum is actually the coefficient of 𝑥. So in this case, it will be positive eight. But also those two factors have to have a product of- in this case it’s negative 20. So it’s last numerical value. So what we want to do for this question is we want to find two factors that give us a sum of positive eight and a product of negative 20.

What I’ve actually written out here just to show us what we’re looking for. I’ve actually included the positive eight — so included the sign there — just to help us remember that we need to be very careful and actually remember to include signs when we’re doing this. Well, we can actually in our question use the products being negative 20 to our advantage cause that’s gonna help us to put actually something else inside our parentheses because if it’s gonna have a product of negative 20, we know that there’s gonna have to be a negative multiplied by a positive because this is the only combination that will actually give us a negative result.

Okay, so great, we now have an 𝑥 in both of our parentheses. We have a positive in one and negative in the other. So all we need to do now is find our values that’re gonna go inside our parentheses. Okay, so now to actually help us find the factors that go inside our parentheses, we’ve listed all the factors of negative 20. So we have negative one multiplied by 20, negative two multiplied by 10, negative four multiplied by five. And then, we’ve put the reverse. So we have one multiplied by negative 20, two multiplied by negative 10, and four multiplied by negative five.

Well, we know these all give us a product of negative 20. But now, let’s actually see what happen if we actually add them together and find out what the sum of them will be and see if it will give us our sum of positive eight. Well, if we see actually the sum of our different factors, we get 19, eight, one, negative 19, negative eight, and negative one.

Well, we’re looking for the pair of factors that give us the sum of positive eight. So therefore, we know the factors we want are negative two and 10 because negative two multiplied by 10 is negative 20 and negative two plus 10 is equal to eight. So therefore, we can say that fully factored 𝑥 squared plus eight 𝑥 minus 20 is equal to 𝑥 plus 10 multiplied by 𝑥 minus two.

Okay, great, so we’ve got our final answer, but what we wanna do now is quickly check. And I’m gonna do that check by expanding our parentheses. So we get 𝑥 multiplied by 𝑥, which gives us 𝑥 squared. 𝑥 multiplied by negative two gives us negative two 𝑥. 10 multiplied by 𝑥 gives us positive 10𝑥. And then, finally, 10 multiplied by negative two, it’s gonna give us negative 20.

So therefore, if we actually collect our like terms, so we got negative two 𝑥 plus 10𝑥 which gives us plus eight 𝑥. We are left with 𝑥 squared plus eight 𝑥 minus 20. So great, this is showing that yes, we’ve got the right factors.

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