Video Transcript
If the roots of the equation two 𝑥 squared plus 10𝑥 plus 12 plus one over 𝑘 equals zero are equal, what is the value of 𝑘?
So, in this question, what we have is a quadratic. And that quadratic is in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero. So, what we can do to evaluate our roots is in fact use our discriminant. And we’re gonna use this because once we evaluate our roots, we’ll be able to then find out the value of 𝑘. Well, the discriminant is found using 𝑏 squared minus four 𝑎𝑐, where 𝑏 is the coefficient of 𝑥, 𝑎 is the coefficient of 𝑥 squared, and 𝑐 is our numerical constant.
And what we know about our discriminant is if it’s greater than zero, so 𝑏 squared minus four 𝑎𝑐 is greater than zero, then our roots are real and different. If the discriminant is equal to zero, then we can say that the roots are real and same. However, if the discriminant is less than zero, then we can say that the roots are not real and they’re complex.
Okay, great. So, we now know how to use our discriminant. But which one is gonna be useful for us in this question? Well, in fact, it’s going to be the middle relationship. 𝑏 squared minus four 𝑎𝑐, when it’s equal to zero, the roots are real and same. And that’s because what we’ve been told in the question is that the roots of our equation are equal. Well, before we can use that relationship, first of all, what we need to do is establish what our 𝑎, 𝑏, and 𝑐 are.
Well, 𝑎 is gonna be equal to two, 𝑏 is equal to 10, and 𝑐 is gonna be equal to 12 plus one over 𝑘. So, therefore, what we can say is that our discriminant, which is 𝑏 squared minus four 𝑎𝑐, is gonna be equal to zero. So, therefore, 10 squared minus four multiplied by two multiplied by 12 plus one over 𝑘 is equal to zero. So, therefore, we can say that 100 minus eight multiplied by 12 plus one over 𝑘 equals zero. So, what we’re gonna do now is distribute across our parentheses.
So, when we do that, we’re gonna get 100 minus 96 minus eight over 𝑘 equals zero. So then, if we subtract 96 from 100, we get four. And then if we add eight over 𝑘 to both sides of the equation, we’re gonna get four is equal to eight over 𝑘. So then, if we multiply both sides of the equation by 𝑘, we get four 𝑘 is equal to eight. So, therefore, 𝑘 is gonna be equal to two, and we get that by dividing each side of the equation by four.
So, we can say that if the roots of the equation two 𝑥 squared plus 10𝑥 plus 12 plus one over 𝑘 equals zero are equal, then the value of 𝑘 is going to be two.
