Suppose the two roots of the equation 𝑥 squared minus 𝑘 plus six 𝑥 minus 10𝑘 minus nine equals zero are equal. Determine all possible values of 𝑘 and then find the two roots.
When we’re looking to solve a problem like this where we have been given information about the roots of a quadratic equation, then what we are going to use is the discriminant. Now, first of all, what we want is an equation in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero. And then what we can say is that the discriminant is 𝑏 squared minus four 𝑎𝑐. And we have three different scenarios for our discriminant. So, what we have is when 𝑏 squared minus four 𝑎𝑐, our discriminant is less than zero; there are no real roots. If 𝑏 squared minus four 𝑎𝑐 is equal to zero, then there is one repeated root. And if 𝑏 squared minus four 𝑎𝑐 is greater than zero, then there are two real and different roots.
Well, in this problem, we’re gonna look at the middle scenario, where 𝑏 squared minus four 𝑎𝑐 is equal to zero, because we’re told the roots are equal. So, therefore, what we have is our one repeated root. So, if we take a look at our equation from the question, then we can see that it’s already in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero. So, what we can do is identify what our 𝑎, 𝑏, and 𝑐 are going to be. Well, because the coefficient of our 𝑥 squared term is one, because it’s just a single 𝑥 squared, 𝑎 is going to be equal to one, then 𝑏 is gonna be equal to negative 𝑘 plus six, and 𝑐 is going to be equal to negative 10𝑘 minus nine.
So, therefore, using the discriminant, we can set up the equation negative 𝑘 minus six all squared ‘cause that’s our 𝑏 squared. And we get that because if we look at 𝑏, it was negative 𝑘 plus six. So, if we distribute across the parentheses, so we multiply each of the terms by the negative one. It gives us our negative 𝑘 minus six. And then we square that minus four multiplied by one multiplied by. Then, we’ve got negative 10𝑘 plus nine equals zero. And again, we got negative 10𝑘 plus nine because what we needed to do is multiply everything in the parentheses, so our 10𝑘 minus nine, by negative one.
So, the first thing we need to do is distribute across the parentheses for negative 𝑘 minus six all squared. When we do that, we get 𝑘 squared plus 12𝑘 plus 36. And we get that because it’s the same as multiplying negative 𝑘 minus six by negative 𝑘 minus six. And if we want to distribute across the parentheses here, first of all, we’d have negative 𝑘 multiplied by negative 𝑘, which gives us 𝑘 squared, and then negative 𝑘 multiplied by negative six, which gives us positive six 𝑘. And then we add another six 𝑘 ‘cause we’ve got negative six multiplied by negative 𝑘 again. And then we add 36 because we’ve got negative six multiplied by negative six. And then collecting like terms gives us our 𝑘 squared plus 12𝑘 plus 36.
Okay, now let’s move on and simplify the rest of our equation. Well, what we’re gonna have is plus 40𝑘. And that’s because we had negative four multiplied by one multiplied by negative 10𝑘, so that’s positive 40𝑘. Then we’ve got minus 36 equals zero. And we got that minus 36 because we had negative four multiplied by positive nine, which gives us negative 36. So when simplified, we’re left with 𝑘 squared plus 52𝑘 equals zero. So, now, to solve, what we can do is factor the left-hand side. And when we do that, we’re gonna have 𝑘 multiplied by 𝑘 plus two equals zero. So, therefore, we can say that 𝑘 is equal to zero or negative 52. And that’s because if we had 𝑘 equal to zero, we’d have zero multiplied by 52, which would just give us zero. And if 𝑘 was equal to negative 52, we’d have negative 52 multiplied by and then we’d have zero because negative 52 plus 52 is zero, which once again would give us our zero.
So, have we finished there? Have we solved the problem? Well, what we have done is found the possible values of 𝑘. So, yes, we’ve solved that part of the problem. However, what we then need to do is find the two roots. So, what we’re going to have to do is substitute our values of 𝑘 back in to our original equation. So, to give us room to do that, what we’re going to do is clear a space on the left-hand side.
So, if we start when 𝑘 is equal to zero, then we’re going to have 𝑥 squared minus six 𝑥. And that’s because we had zero plus six, which is just six. So, minus six 𝑥 plus nine equals zero. And we get plus nine because what we had was negative then we’ve got 10𝑘, which would just be zero, minus nine. So negative multiplied by negative nine, which is gonna give us positive nine. So, then, if we factor once again, what we’re gonna get is 𝑥 minus three multiplied by 𝑥 minus three is equal to zero. And that’s because negative three multiplied by negative three is positive nine and negative three added to negative three is negative six. So, therefore, our roots would just be three or three. So, we could just say that 𝑥 is equal to three. Because if it was three, then that would make each of parentheses equal to zero. So, therefore, the right-hand side would be zero.
Okay, great. That’s when 𝑘 is equal to zero. Now, let’s take a look at when 𝑘 is equal to negative 52. So, when 𝑘 is equal to negative 52, we’re going to have 𝑥 squared minus negative 46𝑥 minus negative 520 minus nine equals zero. So, now if we simplify this, we’re gonna have 𝑥 squared plus 46𝑥 plus 529 equals zero, remembering that we got our positive 46𝑥 and positive 529 because of the negatives in front of our parentheses. So, we had negative multiplied by negative 46, which gave us our positive, and negative multiplied by negative 529, which gave us our positive. So, once again, what we could do now is in fact factor. Well, once again, when we factor, we get repeated root because we have 𝑥 plus 23 multiplied by 𝑥 plus 23 is equal to zero. But this is exactly what we were expecting because the question wanted us to assume the roots of our equation were equal.
So, we have here in both scenarios got roots that are equal, because we can say that when 𝑘 is equal to negative 52, 𝑥 is equal to negative 23. So, therefore, what we could say is that the possible values of 𝑘 and the two roots are 𝑘 equals zero with two roots that both repeated, as we said, three and three, or when 𝑘 equals negative 52, the two repeated roots are negative 23 and negative 23.