Answer:
Our solutions are (-1/2, 1/2) and (-1, 2).
Explanation:
Since both right sides of each equation is equal to y, we can set them equal to each other.
2x² = -3x - 1
Let's get everything on the left side...
2x² + 3x + 1 = 0
Let's solve this quadratic by splitting the middle...
We want two #'s that add up to 3 and multiply to 2. (2×1)
These are 2 and 1. Let's split the middle into 2x and x.
2x² + 2x + x + 1 = 0
Factor the first and last two terms.
2x(x+1)+1(x+1) = 0
(2x+1)(x+1) = 0
If any value of x makes either factor evaluate to 0, it is a solution.
2x + 1 = 0 ⇒ 2x = -1 ⇒ x = -1/2 ⇒ y = 2(-1/2)² = 2(1/4) = 1/2
x + 1 = 0 ⇒ x = -1 ⇒ y = 2(-1)² = 2(1) = 2
Our solutions are (-1/2, 1/2) and (-1, 2).