Solve these equations algebraically. |𝑥^2 − 4| = 4 − 𝑥

Answer 1

To solve this equation algebraically, we will separate it into two cases.

Case 1: 𝑥^2 - 4 = 4 - 𝑥
Rearranging the equation, we get:
𝑥^2 + 𝑥 - 8 = 0
Factoring the quadratic equation, we have:
(𝑥 + 4)(𝑥 - 2) = 0
Setting each factor equal to zero, we get:
𝑥 + 4 = 0 or 𝑥 - 2 = 0
𝑥 = -4 or 𝑥 = 2

Case 2: 𝑥^2 - 4 = 𝑥 - 4
Simplifying the equation, we have:
𝑥^2 - 𝑥 = 0
Factoring out 𝑥, we get:
𝑥(𝑥 - 1) = 0
Setting each factor equal to zero, we get:
𝑥 = 0 or 𝑥 - 1 = 0
𝑥 = 0 or 𝑥 = 1

Therefore, the solution to the equation |𝑥^2 − 4| = 4 − 𝑥 is 𝑥 = -4, 0, 1, or 2.