1. **State the problem:** Solve the equation $\frac{32}{a} = a - 4$ for $a$. 2. **Rewrite the equation:** Multiply both sides by $a$ (assuming $a \neq 0$) to eliminate the denominator: $$32 = a(a - 4)$$ 3. **Expand the right side:** $$32 = a^2 - 4a$$ 4. **Bring all terms to one side to form a quadratic equation:** $$a^2 - 4a - 32 = 0$$ 5. **Use the quadratic formula:** For $ax^2 + bx + c = 0$, solutions are $$a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $a=1$, $b=-4$, $c=-32$. 6. **Calculate the discriminant:** $$\Delta = (-4)^2 - 4 \times 1 \times (-32) = 16 + 128 = 144$$ 7. **Find the roots:** $$a = \frac{4 \pm \sqrt{144}}{2} = \frac{4 \pm 12}{2}$$ 8. **Calculate each root:** - $a = \frac{4 + 12}{2} = \frac{16}{2} = 8$ - $a = \frac{4 - 12}{2} = \frac{-8}{2} = -4$ 9. **Check for restrictions:** Since $a$ is in the denominator in the original equation, $a \neq 0$. Both $8$ and $-4$ are valid. **Final answer:** $$a = 8 \text{ or } a = -4$$