 In this video we provide the solution to question number 19 from the practice final exam for math 1050. We're given a quadratic function f of x equals 2x squared minus 3x plus 2 and we have to set up and simplify the difference quotient f of x plus h minus f of x all over h. So that's where we're going to start with. Take f of x plus h minus f of x and this all sits above h. Our goal is to get rid of h in the denominator. So we have to evaluate f of x plus h. That means everywhere we see an x in the original formula we have to replace it with an x plus h. So we get 2 times x plus h squared minus 3 times x plus h plus 2. Then we subtract from f of x. So we just put down the original formula 2x squared minus 3x plus 2. This all sits above the h from before. We need to expand the f of x plus h part right. So you have that x plus h squared. Foil it out or use the binomial theorem, whichever you prefer. So you'll get 2 times x squared plus 2xh plus h squared. I'm actually going to take the liberty and distribute the 2 right now. So you actually get 2x squared plus 4xh plus 2h squared. Then we have to distribute the negative 3 right here. So it's going to give us negative 3x minus 3h. You have a plus 2 that's in the first group. And then we subtract from it again f of x, which is 2x squared minus 3x plus 2. This all sits above h. Now if we have expanded the f of x plus h correctly, everything from the second group should cancel with something in the first group. So you have a 2x squared minus 2x squared. We're going to have a negative 3x minus negative 3x. And then finally we have a 2 minus 2. All of those cancel out. And so in the numerator what's left behind 4xh plus 2h squared minus 3h all above h. You'll notice that everything in the numerator is now divisible by h. Let's factor it out. That leaves behind 4x plus 2h minus 3. And this all sits above h now. So now we're at the moment of joy and ecstasy here. We found a divisor of h in the numerator. It'll then cancel out with the divisor of h in the denominator. And we're left now with the simplified difference quotient of 4x plus 2h minus 3. This is the quantity we were looking for. And so now we're done.