1. **State the problem:** Solve the quadratic equation $63x^2 = 175$ for $x$. 2. **Rewrite the equation:** To isolate $x^2$, divide both sides by 63: $$x^2 = \frac{175}{63}$$ 3. **Simplify the fraction:** Both numerator and denominator can be divided by 7: $$x^2 = \frac{25}{9}$$ 4. **Take the square root of both sides:** Remember when taking the square root, consider both positive and negative roots: $$x = \pm \sqrt{\frac{25}{9}}$$ 5. **Simplify the square root:** $$x = \pm \frac{\sqrt{25}}{\sqrt{9}} = \pm \frac{5}{3}$$ 6. **Final answer:** $$x = \pm \frac{5}{3}$$