Question Video: Finding the Value of an Unknown in a Quadratic Equation by Using the Relation between Its Coefficient and Its Roots Mathematics

Video Transcript

Given that 𝐿 and 𝐿 squared are the roots of the equation four 𝑥 squared plus 𝑏𝑥 plus 32 equals zero, find the value of 𝑏.

So what we have here is a quadratic equation in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero. And we know that the roots of the equation or solutions are 𝐿 and 𝐿 squared. But how are we gonna use this and the equation we’ve got to help us find out the value of 𝑏? Well, what we have are a couple of relationships to help us because we have some relationships that are to do with 𝑎, 𝑏, and 𝑐, our coefficient of 𝑥 squared, our coefficient of 𝑥, and our numerical term. So, first of all, we know that the sum of the roots is equal to negative 𝑏 over 𝑎. And then we also know that the product of the roots is equal to 𝑐 over 𝑎.

So, first of all, what we can do is to identify our 𝑎, 𝑏, and 𝑐. Well, our 𝑎 is four, our 𝑏 is just 𝑏, and our 𝑐 is 32. Well, first of all, what we’re gonna use is the product of the roots. And we know that the product of the roots is equal to 𝑐 over 𝑎. Well, then what we can say is that the product of the roots, so that’ll be the two roots multiplied together — well, we know the roots are 𝐿 and 𝐿 squared, so it’s 𝐿 multiplied by 𝐿 squared — is going to be equal to our 𝑐, which is 32, over our 𝑎, which is four. Well, therefore, what we’ve got is 𝐿 cubed is equal to eight. So now what we need to do is cube root both sides of the equation. Well, what this is gonna give us is 𝐿 is equal to two. And that’s cause the cube root of 𝐿 cubed is 𝐿 and the cube root of eight is two.

Okay, great. So we now know that one of the roots is two. So now what we need to do is work out the value of 𝑏. But how are we going to do this? Well, what we’re gonna do is move on to our relationship that we know about the sum of the roots. And that is that negative 𝑏 over 𝑎 is the sum of the roots. Well, therefore, what we can say is that 𝐿 plus 𝐿 squared, because that’s gonna be the sum of our roots, is equal to negative 𝑏 over four. Well, we know what the value of 𝐿 is cause we just worked it out. So we’re gonna substitute this in as well.

So when we do this, what we’re gonna have is two plus two squared equals negative 𝑏 over four. Well, two plus two squared is just six. So we got six equals negative 𝑏 over four. So therefore, what we do is multiply both sides of the equation by four and we get 24 is equal to negative 𝑏. So then we divide through by negative one. And what we get is negative 24 is equal to 𝑏. So we arrive at our final answer. And what we can say is that the value of 𝑏 is negative 24.