Aim: How do we solve linear-quadratic systems algebraically?

The way we solve linear-quadratic systems algebraically is by making the two quadratic equations equal to each other.
Example:
  y=2x+8
  y=x^2+x-12
You combine both equations.
  2x+8=x^2+x-12
Then you get both equations to one side by either adding or subtracting to make the whole equation equal to zero.
  0=x^2-x-20
You do the diamond problem to find to multiples that add up to -1 and that multiplies to -20
  0=(x-5)(x+4)
  x=5, x=-4
You then plug back in both numbers for what you got for x but separately into the equations to find what y equals. Then your final answers: (5,18), (-4,0)