Expert: Sherry Wallin - 12/29/2010

Question
y=3x^2-2x-2
x+y=4

Answer
Trisha~
One way to solve this system of equations is to solve each for one variable and it might as well be y since the first one is already solved for y. So the second one is y = 4-x and now set the y's equal to each other:

4-x = 3x^2-2x-2   Now get everything on one side and 0 on the other so you can factor the result.
3x^2-x -6 = 0

Now you can try to factor by sight or you can use the quadratic formula or complete the square to find the roots (solutions mean the same thing). Let's use the quadratic formula. Recall the standard form of a quadratic equation is ax^2+ bx + c = 0. In your case a = 3, b = -1 and c = -6
and the quadratic formula says that x = [-b +-sqrt(b^2-4ac)]/2a, now plug in your numbers:

x = [-(-1) +- sqrt((-1)^2-4(3)(-6))]/2(3)
x = [1 +- sqrt(1+72)]/6
x = [1+- sqrt(73)]/6 ->
x = (1+sqrt(73))/6, (1-sqrt(73))/6

Math Prof

Another way you could have done it is to substitute 3x^2-2x-2 into x + y = 4 for y:
x + 3x^2-2x-2 = 4, move everything to the left leaving 0 on the right: 3x^2 -2 -6 = 0 and proceed as we did in the first method.