1. **State the problem:** Solve the equation $$4 - x \times \frac{x}{23} + \frac{15}{x^2} = 0$$ for $x$. 2. **Rewrite the equation:** The equation is $$4 - \frac{x^2}{23} + \frac{15}{x^2} = 0$$. 3. **Multiply through by $23x^2$ to clear denominators:** $$23x^2 \times 4 - 23x^2 \times \frac{x^2}{23} + 23x^2 \times \frac{15}{x^2} = 23x^2 \times 0$$ 4. **Simplify each term:** $$92x^2 - x^4 + 345 = 0$$ 5. **Rewrite as a standard polynomial:** $$-x^4 + 92x^2 + 345 = 0$$ 6. **Multiply entire equation by $-1$ to simplify:** $$\cancel{-}x^4 + \cancel{-}92x^2 + \cancel{-}345 = 0 \Rightarrow x^4 - 92x^2 - 345 = 0$$ 7. **Substitute $y = x^2$ to reduce degree:** $$y^2 - 92y - 345 = 0$$ 8. **Solve quadratic in $y$ using quadratic formula:** $$y = \frac{92 \pm \sqrt{92^2 - 4 \times 1 \times (-345)}}{2}$$ 9. **Calculate discriminant:** $$92^2 = 8464$$ $$4 \times 345 = 1380$$ $$\sqrt{8464 + 1380} = \sqrt{9844}$$ 10. **Approximate $\sqrt{9844} \approx 99.22$** 11. **Find roots for $y$:** $$y_1 = \frac{92 + 99.22}{2} = \frac{191.22}{2} = 95.61$$ $$y_2 = \frac{92 - 99.22}{2} = \frac{-7.22}{2} = -3.61$$ 12. **Recall $y = x^2$, so $x^2 = 95.61$ or $x^2 = -3.61$** 13. **Discard negative $x^2$ since $x^2 \geq 0$:** $$x^2 = 95.61$$ 14. **Solve for $x$:** $$x = \pm \sqrt{95.61} \approx \pm 9.78$$ **Final answer:** $$x \approx 9.78 \text{ or } x \approx -9.78$$