Video Transcript
Given that the sum of the roots of the equation eight 𝑥 squared plus 𝑏𝑥 plus 18 equals zero is equal to their product, find the value of 𝑏.
So what we have here is a quadratic equation, and it’s in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero. And what we have when we got a quadratic in this form is a couple of really useful relationships. The first one is the sum of the roots is equal to negative 𝑏 over 𝑎 and then the product of roots is equal to 𝑐 over 𝑎. What we’re gonna do is use these useful relationships to help us solve this problem because we’re told that the sum of the roots of the equation is equal to their product.
Well, before we use our relationships to work this out, what we first need to do is identify our 𝑎, 𝑏, and 𝑐. Well, these are eight, 𝑏, and 18. Well, what we’re told is that the sum of the roots is equal to the product of the roots. So therefore, we can say that negative 𝑏 over 𝑎 is equal to 𝑐 over 𝑎. Well, if we substitute in our values, what this will tell us is that negative 𝑏 over eight is equal to 18 over eight. Well then, what we can do is multiply through by eight and what we’re gonna get is negative 𝑏 is equal to 18.
Then, we can divide through by negative one. We’re gonna get 𝑏 is equal to negative 18. So therefore, we can say that given that the sum of the roots of the equation eight 𝑥 squared plus 𝑏𝑥 plus 18 equals zero is equal to their product, then the value of 𝑏 is negative 18.
