Solve Equation 8E63F2

1. **State the problem:** Solve the equation $$x^{2/3} + (8x)^{1/3} = 24$$ for $x$. 2. **Rewrite the equation using substitution:** Let $$y = x^{1/3}$$. Then $$x^{2/3} = (x^{1/3})^2 = y^2$$ and $$(8x)^{1/3} = 8^{1/3} x^{1/3} = 2y$$ because $8^{1/3} = 2$. 3. **Substitute into the equation:** $$y^2 + 2y = 24$$ 4. **Rewrite as a quadratic equation:** $$y^2 + 2y - 24 = 0$$ 5. **Solve the quadratic equation:** Use the quadratic formula $$y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=2$, and $c=-24$. Calculate the discriminant: $$\Delta = 2^2 - 4 \times 1 \times (-24) = 4 + 96 = 100$$ Calculate the roots: $$y = \frac{-2 \pm \sqrt{100}}{2} = \frac{-2 \pm 10}{2}$$ So, $$y_1 = \frac{-2 + 10}{2} = \frac{8}{2} = 4$$ $$y_2 = \frac{-2 - 10}{2} = \frac{-12}{2} = -6$$ 6. **Back-substitute to find $x$:** Recall $$y = x^{1/3}$$, so $$x = y^3$$ For $$y_1 = 4$$: $$x = 4^3 = 64$$ For $$y_2 = -6$$: $$x = (-6)^3 = -216$$ 7. **Check solutions in the original equation:** - For $$x=64$$: $$x^{2/3} = (64)^{2/3} = ((64)^{1/3})^2 = 4^2 = 16$$ $$(8x)^{1/3} = (8 \times 64)^{1/3} = (512)^{1/3} = 8$$ Sum: $$16 + 8 = 24$$ which satisfies the equation. - For $$x=-216$$: $$x^{2/3} = ((-216)^{1/3})^2 = (-6)^2 = 36$$ $$(8x)^{1/3} = (8 \times -216)^{1/3} = (-1728)^{1/3} = -12$$ Sum: $$36 + (-12) = 24$$ which also satisfies the equation. **Final answer:** $$x = 64 \text{ or } x = -216$$