Solve the Absolute Value Quadratic: |x²-11|=5

To solve the equation $|x^2 - 11| = 5$, we will consider the two cases implied by the property of absolute values:

• Case 1: $x^2 - 11 = 5$
• Case 2: $x^2 - 11 = -5$

Let's solve each case separately.

Case 1: $x^2 - 11 = 5$
First, add 11 to both sides: $x^2 = 16$.
Taking the square root of both sides, we get $x = \pm\sqrt{16} = \pm4$.
Thus, the solutions for Case 1 are $x = 4$ and $x = -4$.

Case 2: $x^2 - 11 = -5$
First, add 11 to both sides: $x^2 = 6$.
Taking the square root of both sides, we get $x = \pm\sqrt{6}$.
Thus, the solutions for Case 2 are $x = \sqrt{6}$ and $x = -\sqrt{6}$.

Therefore, combining solutions from both cases, the complete set of solutions is $x = 4, -4, \sqrt{6}, -\sqrt{6}$.

Reviewing the multiple-choice options, both cases' solutions correspond to the correct answer listed as 'Answers a + b.'