I don't know how to solve y=-x^2 + 14x + - 24

What numbers multiply to get 24 and add to get 14?

6 * 4 = 24; 6 + 4 = / = 14

12 * 2 = 24; 12 + 2 = 14! Yay!

Let's split 14 up, then.

y = - x^2 + 2x + 12x - 24

Now we can just factor by grouping!

-x (x-2) + 12 (x-2)

Now that we have our factored problem (-x+12) (x-2) = y we can plug in 0 for y and see what we get. This is because of the zero product property - if one number is zero, then they will multiply to get zero.

Now we can just solve like two normal equations.

-x + 12 = 0 or x - 2 = 0

- 12 - 12 + 2 + 2

-x = - 12 or x = 2

x = { 12, 2 }