Video Transcript
If the roots of the equation 𝑥 squared minus eight multiplied by 𝑘 plus one 𝑥 plus 64 equals zero are equal, find the possible values of 𝑘.
Well, in this question, what we’re going to do is use the discriminant to help us solve the problem, because the discriminant tells us about the roots of a quadratic equation. Now, if we have a quadratic equation in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero, then what the discriminant tells us is that if 𝑏 squared minus four 𝑎𝑐, so our discriminant, is less than zero, then there are no real roots. If 𝑏 squared minus four 𝑎𝑐 is equal to zero, there is one repeated root. And if 𝑏 squared minus four 𝑎𝑐, our discriminant, is greater than zero, there are two different and real roots.
So as our equation is already in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero, then what we can do is identify what our 𝑎, 𝑏, and 𝑐 are going to be. Our 𝑎 issss going to be one. And that’s because the coefficient of 𝑥 squared is one, because there isn’t a coefficient, so that means there’s just a single 𝑥 squared. Our 𝑏, which is the coefficient of 𝑥, is going to be negative eight multiplied by 𝑘 plus one, remembering here to include the sign. And then our 𝑐 is going to be 64.
Well, now, to set up an equation, what we do is we have a look at which form of the discriminant we’re interested in. And that is where 𝑏 squared minus four 𝑎𝑐 is equal to zero, because in the question we’re asked to find the possible values of 𝑘 if the roots of the equation are equal. So using this, what we have is negative eight multiplied by 𝑘 plus one all squared minus four multiplied by one multiplied by 64 is equal to zero. And then distributing across the parentheses, we’re gonna get negative eight 𝑘 minus eight all squared minus 256 equals zero.
So now just to remind ourselves what we do if we have negative eight 𝑘 minus eight all squared, which is gonna show how it works. So it’s the same as negative eight 𝑘 minus eight multiplied by negative eight 𝑘 minus eight. And then what we do is distribute across the parentheses. So we multiply each of the terms in the first parentheses by each of the terms in the second. So we start with negative eight 𝑘 multiplied by negative eight 𝑘, which is gonna give us 64𝑘 squared. And then we have negative eight 𝑘 multiplied by negative eight, which is gonna give us positive 64𝑘. And then we add another 64𝑘 because we’ve got negative eight multiplied by negative eight 𝑘 once again. And then, finally, we have negative eight multiplied by negative eight, which gives us positive 64. And then simplifying like terms, we get 64𝑘 squared plus 128𝑘 plus 64.
Okay, so let’s get back and solve the problem to find 𝑘. So what we’re going to have is 64𝑘 squared plus 128𝑘 plus 64 minus 256 is equal to zero. And this will simplify to 64𝑘 squared plus 128𝑘 minus 192 equals zero. And then what we can see is that to make this easier to solve, we can actually divide through by 64, which is gonna give us 𝑘 squared plus two 𝑘 minus three equals zero.
So now what we do is we factor to solve. And when we do that, we get 𝑘 plus three multiplied by 𝑘 minus one equals zero. So therefore, when we solve, we can say that 𝑘 is equal to negative three or one. And that’s because these are the values that would make either of our parentheses equal to zero, because then if we have zero multiplied by anything, we get zero. So, for instance, we add negative three plus three; that’s zero. So zero multiplied by anything, whatever is in the second parentheses, would be equal to zero.
So therefore, we can say that if the roots of the equation 𝑥 squared minus eight multiplied by 𝑘 plus one 𝑥 plus 64 equals zero are equal, then the possible values of 𝑘 are negative three or one.
