> k12< on a math test, amy was asked to find the point(s) where the equations y = x2 + 3x – 5 & y = 4x + 1 intersect. amy’s solution is shown below. amy’s solution: y = x2 + 3x – 5 y = 4x + 1 x2 + 3x – 5 = 4x + 1 x2 – x – 6 = 0 (x – 3) (x + 2) = 0 x = 3 or x = -2 the graphs intersect at (-2, 0) and (3, 0) a) is amy’s solution correct? explain why or why not. b) if amy’s solution is incorrect, show the correct work for this problem.

The correct answer is a) Amy's solutions are correct.

Since she wanted the intersection point of the equations, she had to set them equal.  Once she did she had

x²+3x-5=4x+1

Gathering the variables on one side of the equation, she subtracted 4x from each side:
x²+3x-5-4x=4x+1-4x
x²-x-5=1

Quadratics should be equal to 0 when we solve them, so she subtracted 1 from each side:
x²-x-5-1=1-1
x²-x-6=0

She then factored her expression; factors of -6 that sum to -1 are -3 and 2:
(x-3)(x+2)=0

Using the zero product property we know that either x-3=0 or x+2=0; therefore x=3 or x=-2.