If the sum of the roots of the equation negative three 𝑥 squared plus 𝑘𝑥 plus 11 equals zero is four, what is value of 𝑘?
So, what we have in this question is a quadratic that’s in a nice useful form cause it’s in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero. But why is this useful? Well, it’s useful because what we have are a couple of relationships relating to the roots of an equation. First of all, the sum of the roots are equal to negative 𝑏 over 𝑎. And the product of the roots are equal to 𝑐 over 𝑎. And we see that we’ve got our 𝑎, 𝑏, and 𝑐 when we have our quadratic in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero.
Well, the first thing we always do is identify our 𝑎, 𝑏, and 𝑐. So, we have our negative three for our 𝑎, 𝑘 for our 𝑏, and 11 for our 𝑐. It’s worth mentioning at this point that the sign is very important. Hence, for 𝑎, we’ve got negative three not just three. Okay, we’ve got a, 𝑏, and 𝑐. What’s next?
Well, we’re told in the question that the sum of the roots is four. So therefore, we’re gonna use the relationship that tells us that sum of the roots is equal to negative 𝑏 over 𝑎. So therefore, what we’re gonna have when we substitute in our values is four, cause that’s our sum, is equal to negative 𝑘 over negative three. Well, we know that a negative divided by a negative is a positive, so we can rewrite negative 𝑘 over negative three as just 𝑘 over three.
So, great, we’re now in a position where we can find 𝑘 quite easily because all we need to do is multiply both sides of our equation by three. And when we do that, what we’re gonna get is 12 is equal to 𝑘.
So therefore, what we can say is that if the sum of the roots of the equation negative three 𝑥 squared plus 𝑘𝑥 plus 11 equals zero is four, then the value of 𝑘 is 12.