Given that negative one is one of the roots of the equation 𝑥 squared plus 𝑎𝑥 plus two equals zero, find the value of 𝑎 and the value of the other root.
So what we have here is a quadratic equation 𝑥 squared plus 𝑎𝑥 plus two equals zero. Well, the key bit of information from this problem is that we’re told that negative one is one of the roots of the equation. So therefore, what we know is the factored form of the equation must be 𝑥 plus one multiplied by 𝑥 plus another number, which we’re gonna call 𝑏, is equal to zero. And we know that the first parentheses is 𝑥 plus one because we’re told that negative one is one of the roots. So therefore, that means that the 𝑥-value of negative one must make one of the parentheses equal to zero and negative one add one is equal to zero.
So now what we’re gonna do is distribute across our parentheses. So what we’re gonna do is multiply both terms in the first set of parentheses by both terms in the second set of parentheses. So first of all, we’ve got 𝑥 multiplied by 𝑥, which is 𝑥 squared. Then we have 𝑥 multiplied by 𝑏, which is 𝑏𝑥. Then we’re gonna move on to the positive one. So we have positive one multiplied by 𝑥, which is just 𝑥, and then, finally, one multiplied by 𝑏, which gives us 𝑏. Okay, great. So we now have our quadratic 𝑥 squared plus 𝑏𝑥 plus 𝑥 plus 𝑏. Well, if we take a look at our original quadratic, we can see that the numerical value on the end, so the number without any 𝑥-term, is positive two. So therefore, this is what 𝑏 must be equal to because 𝑏 on its own is the only non-𝑥-term that we have.
Well, as we’ve worked out that 𝑏 is equal to two, then this tells us that the other root must be negative two. And that’s because, for the same reasons we gave before, if we want to make the right parentheses equal to zero, then if 𝑏 is equal two, then the 𝑥-value that will make that equals zero is negative two because negative two add two is zero. So now there’s just one more part of the question left to complete, and that is to find the value of 𝑎.
Well, if we substitute in our 𝑏-value of two, what we’re gonna get is 𝑥 squared plus two 𝑥 plus 𝑥 plus two equals zero. So now what we can is collect like terms. And when we do that, we’re left with a quadratic 𝑥 squared plus three 𝑥 plus two equals zero. Well, as 𝑎 is the coefficient of 𝑥 and in our new quadratic the coefficient of 𝑥 is three, we can therefore say that 𝑎 is equal to three. So therefore, we found the value of the other root, which is negative two, and the value of 𝑎, which is three.