Without solving the equation negative three 𝑥 squared minus 16𝑥 plus 63 equals zero, find the sum of its roots.
So what we have here is a quadratic equation in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero. And what we’re told to do is find the sum of its roots. Well, instinct might think, well, let’s solve this and then we’ll be able to find each root and add them together. However, the question states it does not want us to solve the equation. So how else can we do this? Well, what we actually have are a couple of formulae that can help us. And they can help us by using what we have for our coefficients of our 𝑥 terms and also the numerical term in our quadratic.
First of all, we know that the sum of roots is equal to negative 𝑏 over 𝑎. We also know the product of the roots is equal to 𝑐 over 𝑎. We don’t need this for this question, but it was worth having to look at. So what we’re gonna use is the first one, the sum of the roots is equal to negative 𝑏 over 𝑎. Well, we can see from our quadratic that our 𝑎, 𝑏, and 𝑐 are negative three, negative 16, and 63, respectively. And what we need to do is remember the signs which we’ve done here. So we’ve got two negatives and a positive.
Now, we only need 𝑎 and 𝑏 to find out the sum of the roots. So let’s plug these in. Well, when we plug them in, what we’re gonna get is the sum of the roots is equal to negative negative 16 over negative three. Well, negative negative 16 is gonna give us positive 16. Well, positive 16 over negative three gives us negative 16 over three. So we can say this is the sum of the roots of the equation negative three 𝑥 squared minus 16𝑥 plus 63 equals zero.