Questions: Solve the absolute value equation. 𝑥²−8=5

Solution Steps

To solve the absolute value equation \(|x^2 - 8| = 5\), we need to consider two separate cases: \(x^2 - 8 = 5\) and \(x^2 - 8 = -5\). Solving these equations will give us the possible values of \(x\).

We start with the equation from the absolute value: \[ x^2 - 8 = 5 \] Rearranging gives: \[ x^2 = 13 \] Taking the square root of both sides, we find: \[ x = \pm \sqrt{13} \approx \pm 3.6056 \]

Next, we consider the second case: \[ x^2 - 8 = -5 \] Rearranging gives: \[ x^2 = 3 \] Taking the square root of both sides, we find: \[ x = \pm \sqrt{3} \approx \pm 1.7321 \]

Combining the solutions from both cases, we have: \[ x \approx 3.6056, -3.6056, 1.7321, -1.7321 \]

Final Answer