1. The problem is to solve the equation $$1499.744 = \frac{1}{3} \times 2 \times \left[a^2 + 16a + 64 + a^2 + \sqrt{a^2 \times (a + 8)^2}\right]$$ for $a$. 2. Simplify the constants and expressions inside the brackets: $$\frac{1}{3} \times 2 = \frac{2}{3}$$ Inside the brackets, combine like terms: $$a^2 + 16a + 64 + a^2 = 2a^2 + 16a + 64$$ Also, simplify the square root term: $$\sqrt{a^2 \times (a + 8)^2} = \sqrt{(a(a+8))^2} = |a(a+8)|$$ 3. Since $a(a+8)$ is squared inside the root, the root is the absolute value: $$|a(a+8)|$$ Assuming $a$ is such that $a(a+8) \geq 0$, then: $$|a(a+8)| = a(a+8) = a^2 + 8a$$ 4. Substitute back: $$1499.744 = \frac{2}{3} \times \left(2a^2 + 16a + 64 + a^2 + 8a\right) = \frac{2}{3} \times \left(3a^2 + 24a + 64\right)$$ 5. Multiply both sides by $\frac{3}{2}$ to isolate the quadratic expression: $$1499.744 \times \frac{3}{2} = 3a^2 + 24a + 64$$ Calculate left side: $$1499.744 \times 1.5 = 2249.616$$ 6. Write the quadratic equation: $$3a^2 + 24a + 64 = 2249.616$$ Move all terms to one side: $$3a^2 + 24a + 64 - 2249.616 = 0$$ $$3a^2 + 24a - 2185.616 = 0$$ 7. Divide entire equation by 3 to simplify: $$a^2 + 8a - 728.5387 = 0$$ 8. Use the quadratic formula: $$a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=8$, $c=-728.5387$. Calculate discriminant: $$\Delta = 8^2 - 4 \times 1 \times (-728.5387) = 64 + 2914.1548 = 2978.1548$$ 9. Calculate square root of discriminant: $$\sqrt{2978.1548} \approx 54.58$$ 10. Calculate roots: $$a = \frac{-8 \pm 54.58}{2}$$ 11. First root: $$a = \frac{-8 + 54.58}{2} = \frac{46.58}{2} = 23.29$$ 12. Second root: $$a = \frac{-8 - 54.58}{2} = \frac{-62.58}{2} = -31.29$$ 13. Check the assumption $a(a+8) \geq 0$: - For $a=23.29$, $23.29 \times (23.29 + 8) > 0$ true. - For $a=-31.29$, $-31.29 \times (-31.29 + 8) = -31.29 \times (-23.29) > 0$ true. Both roots satisfy the assumption. Final answer: $$a \approx 23.29 \text{ or } a \approx -31.29$$