Topic B: Lessons 14-15: The quadratic formula
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Use the quadratic formula to solve the equation, negative x squared plus 8x is equal to 1. Now, in order to really use the quadratic equation, or to figure out what our a's, b's and c's are, we have to have our equation in the form, ax squared plus bx plus c is equal to 0. And then, if we know our a's, b's, and c's, we will say that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac-- all of that over 2a. So the first thing we have to do for this equation right here is to put it in this form. And on one side of this equation, we have a negative x squared plus 8x, so that looks like the first two terms. But our constant is on the other side. So let's get the constant on the left hand side and get a 0 here on the right hand side. So let's subtract 1 from both sides of this equation. The left hand side of the equation will become negative x squared plus 8x minus 1. And then the right hand side, 1 minus 1 is 0. Now we have it in that form. We have ax squared a is negative 1. So let me write this down. a is equal to negative 1. a is equal to negative 1. It's implicit there, you could put a 1 here if you like. A negative 1. Negative x squared is the same thing as negative 1x squared. b is equal to 8. So b is equal to 8, that's the 8 right there. And c is equal to negative 1. That's the negative 1 right there. So now we can just apply the quadratic formula. The solutions to this equation are x is equal to negative b. Plus or minus the square root of b squared, of 8 squared, minus 4ac-- let me do it in that green color --minus 4, the green is the part of the formula. The colored parts are the things that we're substituting into the formula. Minus 4 times a, which is negative 1, times negative 1, times c, which is also negative 1. And then all of that-- let me extend the square root sign a little bit further --all of that is going to be over 2 times a. In this case a is negative 1. So let's simplify this. So this becomes negative 8, this is negative 8, plus or minus the square root of 8 squared is 64. And then you have a negative 1 times a negative 1, these just cancel out just to be a 1. So it's 64 minus is 4. That's just that 4 over there. All of that over negative 2. So this is equal to negative 8 plus or minus the square root of 60. All of that over negative 2. And let's see if we can simplify the radical expression here, the square root of 60. Let's see, 60 is equal to 2 times 30. 30 is equal to 2 times 15. And then 15 is 3 times 5. So we do have a perfect square here. We do have a 2 times 2 in there. It is 2 times 2 times 15, or 4 times 15. So we could write, the square root of 60 is equal to the square root of 4 times the square root of 15, right? The square root of 4 times the square root of 15, that's what 60 is. 4 times 15. And so this is equal to-- square root of 4 is 2 times the square of 15. So we can rewrite this expression, right here, as being equal to negative 8 plus or minus 2 times the square root of 15, all of that over negative 2. Now both of these terms right here are divisible by either 2 or negative 2. So let's divide it. So we have negative 8 divided by negative 2, which is positive 4. So let me write it over here. Negative 8 divided by negative 2 is positive 4. And then you have this weird thing. Plus or minus 2 divided by negative 2. And really what we have here is 2 expressions. But if we're plus 2 and we divide by negative 2, it will be negative 1. And if we take negative 2 and divide by negative 2, we're going to have positive 1. So instead of plus or minus, you could imagine it is going to be minus or plus. But it's really the same thing. Right? It's really now minus or plus. If it was plus, it's now going to be a minus. If it was a minus, it's now going to be a plus. Minus or plus 2 times the square root of 15. Or another way to view it is that the two solutions here are 4 minus two roots of 15, and 4 plus two roots of 15. These are both values of x that'll satisfy this equation. And if this confuses you, what I did, turning a plus or minus into minus plus. Let me just take a little bit of an aside there. I could write this expression up here as two expressions. That's what the plus or minus really is. There's a negative 8 plus 2 roots of 15 over negative 2. And then there's a negative 8 minus 2 roots of 15 over negative 2. This one simplifies to-- negative 8 divided by negative 2 is 4. 2 divided by negative 2 is negative 1. 2 times a 4 minus the square root of 15. And then over here you have negative 8 divided by negative 2, which is 4. And then negative 2 divided by negative 2, which is plus the square of 15. And I just realized I made a mistake up here. When we're dividing a 2 divided by negative 2, we don't have this 2 over here. This is just a plus or minus the root of 15. We just saw that when I did it out here. So this is minus the square root of 15. And this is plus the square root of 15. So the two solutions for this equation-- It's good that I took that little hiatus there, that little aside there. The two solutions could be 4 minus the square root of 15, or x, or and, x could be 4 plus the square root of 15. Either of those values of x will satisfy this original quadratic equation.