 The second method for solving quadratic equations that we're going to focus on is the quadratic formula. Many of you have probably dealt with the quadratic formula in the past, so I'm going to just write it out here at the top in order to remind you of it. So the key is you need to remember this first part. If ax squared plus bx plus c equals zero, then x equals negative b plus or minus the square root of b squared minus 4ac all over 2a. Most people remember just that second part of the quadratic formula forgetting that the equation has to first be equal to zero in order for the quadratic formula to work. As most of you have used the quadratic formula in the past, I'm just going to go through one example in order to help you review how to use this. The example we're going to look at today is 4x squared equals 12x minus 15. The first thing you need to make sure to do is set the equation equal to zero. In this case, I'm going to move everything to the left side of the equation just so that my a value remains positive. Now looking at this equation, since it is equal to zero, we see that a is equal to 4, b is equal to negative 12, and c is equal to 15. So I now can plug it into the quadratic formula. Feel free to go back and look at the formula if you need to. It starts with negative b. So our negative of negative 12 will be a positive 12 plus or minus the square root of b squared, which in our case is negative 12 squared minus 4 times a, a is 4 times c, c is 15. And this entire part is divided by 2a, in this case 2 times 4. I think the easiest way to evaluate this expression is to first simplify the part underneath the square root. Negative 12 squared is 144, 4 times 4 times 15 is 240, and all of that will be divided by 8. If we simplify the square root further, we find that we're taking the square root of negative 96. Now I'm going to go to a new slide just to continue to simplify this expression. Notice here we're taking the square root of a negative. Any time the part underneath the square root is negative, that's called the discriminant, we know we're going to get imaginary solutions, because the square root of a negative always introduces an i. I typically just rewrite the expression like this, 12 plus or minus i square root of 96, and still over 8. Now I'm just going to talk about finding an approximate solution, not worrying so much about the exact solution and the exact simplification. So what I suggest doing is splitting up the expression into the real part, 12 over 8, and the imaginary part, i square root of 96 over 8. And then if we're just looking for an approximation, you can divide 12 divided by 8, gives you 1.5, and then here I just would type this into my calculator. The square root of 96 divided by 8 gives us an approximation of 1.225i. For me that's fine if we just do an approximation at this point. We can look further into getting exact values in class if you want.