Given that the roots of the equation four 𝑥 squared minus 12𝑥 plus 𝑘 equals zero are real and different, find the interval which contains 𝑘.
So in this problem, we know that we’re going to use the discriminant to help us solve it. And that’s because what we are told is some information about the roots. We are told that the roots of the equation are real and different. And we can use the discriminant when we have a quadratic equation in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero. And when we have that, the discriminant is 𝑏 squared minus four 𝑎𝑐. And what it in fact does is it tells us some of the conditions for our roots.
And the three conditions that we have for our roots are, first of all, 𝑏 squared minus four 𝑎𝑐 is less than zero. So our discriminant is less than zero. So this means that we’ll have no real roots. If 𝑏 squared minus four 𝑎𝑐 is equal to zero, we have a repeated root. And if 𝑏 squared minus four 𝑎𝑐 is greater than zero, then our roots are real and different.
Well, for this problem, we’re interested in the third situation, where our discriminant is greater than zero, because what we’re looking for are roots that are real and different. Well, for our problem, we already have the quadratic equation in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero. So what we can do is identify our 𝑎, 𝑏, and 𝑐. So our 𝑎, which is the 𝑥 squared coefficient, is going to be four; our 𝑏 is negative 12; and our 𝑐 is going to be equal to positive 𝑘.
So therefore, as we know that the roots are real and different, what we can do is set up an inequality. And that inequality is negative 12 all squared minus four multiplied by four multiplied by 𝑘 is greater than zero. So therefore, this is going to tell us that 144 minus 16𝑘 is greater than zero. So then what we can do with the inequality is add 16𝑘 to each side. And when we do that, we get 144 is greater than 16𝑘. And then if we divide through by 16, what we’re going to get is nine is greater than 𝑘. Well, as we have the inequality nine is greater than 𝑘, what this means is that 𝑘 can take any value less than nine.
So therefore, we can say that the interval which contains 𝑘 is between negative infinity and nine but not including negative infinity or nine. And we’ve shown it here with our interval notation because we’ve used parentheses because we use parentheses when it means that it’s not including that value. However, if we’d used squared brackets, then this would’ve meant that it could include that value as well.