1. Stating the problem: Solve the equation $$x^4 - 30x^2 + 125 = 0$$ for $x$. 2. Recognize this is a quartic equation but can be treated as a quadratic in terms of $x^2$. 3. Let $y = x^2$. Then the equation becomes: $$y^2 - 30y + 125 = 0$$ 4. Use the quadratic formula to solve for $y$: $$y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-30$, and $c=125$. 5. Calculate the discriminant: $$\Delta = (-30)^2 - 4 \times 1 \times 125 = 900 - 500 = 400$$ 6. Find the roots for $y$: $$y = \frac{30 \pm \sqrt{400}}{2} = \frac{30 \pm 20}{2}$$ 7. Calculate each root: - For $+$ sign: $$y = \frac{30 + 20}{2} = \frac{50}{2} = 25$$ - For $-$ sign: $$y = \frac{30 - 20}{2} = \frac{10}{2} = 5$$ 8. Recall $y = x^2$, so solve for $x$: - When $x^2 = 25$, $$x = \pm \sqrt{25} = \pm 5$$ - When $x^2 = 5$, $$x = \pm \sqrt{5}$$ 9. Final solution set: $$x = \pm 5, \pm \sqrt{5}$$