Video Transcript
If the roots of the equation four 𝑥 squared minus 𝑘𝑥 plus one equals zero are equal, what are the possible values of 𝑘?
So what we have here in this question is a quadratic. And it’s in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero. So the good thing about having our equation in this form, so a quadratic in this form, is that we can use the discriminant to help us evaluate the roots. And the discriminant is found using 𝑏 squared minus four 𝑎𝑐, where 𝑎 is not zero. But we’ve said this is gonna help us evaluate the roots, but how?
Well, it’s because we have a few relationships to do with our discriminant that can help us. First of all, if the discriminant 𝑏 squared minus four 𝑎𝑐 is greater than zero, then we know that our roots are real and different. However, if our discriminant is equal to zero, then we know that our roots are real and the same. So what this means is, in practice, is that we have a repeated root. And finally, if the discriminant is less than zero, then what we know about the roots is that they’re not real and they’re complex.
So therefore, the relationship that we’re gonna be interested in in this question is the second one, 𝑏 squared minus four 𝑎𝑐. So our discriminant is equal to zero. And that’s because we’re told that in the question our roots to the equation are equal. So the first thing we want to do is identify our 𝑎, 𝑏, and 𝑐 in the equation we’re looking at.
Well, our 𝑎 is gonna be equal to four, our 𝑏 is gonna be equal to negative 𝑘, and our 𝑐 is gonna be equal to one. And it’s important to remember to include our signs. So that’s why we got negative 𝑘, not just 𝑘. So therefore, what we’re gonna get is our discriminant, which gonna be negative 𝑘 all squared minus four multiplied by four multiplied by one is equal to zero. And the reason we can set up this equation is because we’re using the middle relationship that we already pointed out. And that was because we know that our roots are equal, so they are real and the same.
So what this is gonna give us is 𝑘 squared minus 16 equals zero. So then if we add 16 to both sides of the equation, we’re gonna get 𝑘 squared is equal to 16. Well then, if we take the square root of both sides of the equation, we’re gonna get 𝑘 is equal to positive or negative four. And we get two results because both positive four and negative four when squared will give us 16.
So therefore, if the roots of the equation four 𝑥 squared minus 𝑘𝑥 plus one equals zero are equal, then the possible values of 𝑘 are four and negative four.
