1. **State the problem:** Solve for $y$ in the equation $x^2 = y^2 + 8$. 2. **Rewrite the equation:** We want to isolate $y^2$, so subtract 8 from both sides: $$x^2 - 8 = y^2$$ 3. **Take the square root:** To solve for $y$, take the square root of both sides: $$y = \pm \sqrt{x^2 - 8}$$ 4. **Important note:** The expression under the square root, called the radicand, must be non-negative for real $y$ values. So: $$x^2 - 8 \geq 0 \implies x^2 \geq 8 \implies |x| \geq \sqrt{8} = 2\sqrt{2}$$ 5. **Final answer:** $$y = \pm \sqrt{x^2 - 8} \quad \text{for} \quad |x| \geq 2\sqrt{2}$$