1. **State the problem:** Solve the quadratic equation $$5(x^2 - 7) - 1 = 9$$ for all values of $x$ in simplest form. 2. **Expand and simplify:** Distribute the 5 inside the parentheses: $$5x^2 - 35 - 1 = 9$$ Combine like terms on the left: $$5x^2 - 36 = 9$$ 3. **Isolate the quadratic term:** Add 36 to both sides: $$5x^2 - 36 + 36 = 9 + 36$$ $$5x^2 = 45$$ 4. **Divide both sides by 5:** $$\cancel{5}x^2 = \frac{45}{\cancel{5}}$$ $$x^2 = 9$$ 5. **Solve for $x$ by taking the square root:** $$x = \pm \sqrt{9}$$ $$x = \pm 3$$ **Final answer:** $$x = 3 \text{ or } x = -3$$