If the roots of the equation five 𝑥 squared minus two 𝑘𝑥 plus five equals zero are equal, what are the possible values of 𝑘?
So, what we have here is a quadratic. And with a quadratic, what we can do is use the discriminant to help us decide whether our roots are gonna be equal, whether they’re gonna have real roots, or they’re gonna have no real roots. So, the discriminant is 𝑏 squared minus four 𝑎𝑐. But what does this mean? What’s 𝑏, what’s 𝑎, and what’s 𝑐? Well, 𝑎𝑏 and 𝑐 are parts of our quadratic when we have it in the form that we have. So, we’ve got 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐. So, 𝑎 is the coefficient of 𝑥 squared, 𝑏 is the coefficient of 𝑥, and 𝑐 is our constant term, or our numerical value.
Well, with our quadratic, we’ve got 𝑎 equals five. And that’s because that’s our coefficient of 𝑥 squared. 𝑏 is equal to negative two 𝑘. And be very careful here. Make sure that we include the negative, so the sign. And then, our 𝑐 is gonna be equal to five. But how are these and our discriminant useful? Well, the discriminant is useful because it can tell us how many roots our quadratic will have. So, for instance, if we have 𝑏 squared minus four 𝑎𝑐 is less than zero, then there are no real roots. If 𝑏 squared minus four 𝑎𝑐 is greater than zero, then there are real roots. And if 𝑏 squared minus four 𝑎𝑐 equals zero, then the roots are going to be equal.
And I’ve added some sketches to show what the graphs would look like. So, we can see if it has no roots, that it wouldn’t cross the 𝑥-axis at all. If there are real roots, then it’ll cross our 𝑥-axis in two distinct places. And if the roots were equal, it would just touch the 𝑥-axis. Okay, great, so, we now know what the discriminant is. And we know how it could be used. Let’s use it to solve our problem.
Well, in our question, we’re told that the roots are equal. So therefore, it’s this third scenario we’re interested in when 𝑏 squared minus four 𝑎𝑐 is equal to zero. So therefore, if we substitute in our values 𝑎, 𝑏, and 𝑐, we can have negative two 𝑘 all squared, that’s because that’s our 𝑏, minus four multiplied by 𝑎, which is five, multiplied by five, which is 𝑐. And then, this is all equal to zero because, as we said, we have equal roots.
So, what we’re gonna get is four 𝑘 squared minus 100 equals zero. So, then, if we add 100 to each side of the equation, we’re gonna get four 𝑘 squared is equal to 100. And then, what we need to do is divide both sides of the equation by four cause we want to find out what 𝑘 is going to be. So, we get 𝑘 squared is equal to 25. So, then, if we take the square root of both sides of the equation, we’re gonna get 𝑘 is equal to positive or negative five.
And that’s because if you square root 𝑘 squared, you get 𝑘. And you square root 25, we’re gonna get positive or negative five. That’s because five multiplied by five gives us 25, and negative five multiplied by negative five gives us 25. So therefore, given the roots of the equation five 𝑥 squared minus two 𝑘𝑥 plus five equals zero are equal, the possible values of 𝑘 are five and negative five.