If $3x + 2y = 14$ and $xy = 8$, then $(27x^3 + 8y^3)$ is equal to :

Question

Accepted Answer

To solve the problem and verify why Option 2 is the correct answer, we need to find the value of $27x^3 + 8y^3$ given the equations $3x + 2y = 14$ and $xy = 8$.

### Step-by-Step Explanation:

1. **Given Equations:**
   - $3x + 2y = 14$
   - $xy = 8$

2. **Express $x$ and $y$ in terms of a single variable:**
   - From $3x + 2y = 14$, express $y$ in terms of $x$:
     $$
     2y = 14 - 3x \implies y = \frac{14 - 3x}{2}
     $$

3. **Substitute $y$ in $xy = 8$:**
   $$
   x \left(\frac{14 - 3x}{2}\right) = 8
   $$
   $$
   \frac{x(14 - 3x)}{2} = 8
   $$
   $$
   x(14 - 3x) = 16
   $$
   $$
   14x - 3x^2 = 16
   $$
   $$
   3x^2 - 14x + 16 = 0
   $$

4. **Solve the quadratic equation:**
   - The quadratic equation is $3x^2 - 14x + 16 = 0$.
   - Using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 3$, $b = -14$, $c = 16$:
     $$
     x = \frac{14 \pm \sqrt{(-14)^2 - 4 \cdot 3 \cdot 16}}{6}
     $$
     $$
     x = \frac{14 \pm \sqrt{196 - 192}}{6}
     $$
     $$
     x = \frac{14 \pm \sqrt{4}}{6}
     $$
     $$
     x = \frac{14 \pm 2}{6}
     $$
     $$
     x = \frac{16}{6} \quad \text{or} \quad x = \frac{12}{6}
     $$
     $$
     x = \frac{8}{3} \quad \text{or} \quad x = 2
     $$

5. **Find corresponding $y$ values:**
   - If $x = \frac{8}{3}$, then:
     $$
     y = \frac{14 - 3 \times \frac{8}{3}}{2} = \frac{14 - 8}{2} = 3
     $$
   - If $x = 2$, then:
     $$
     y = \frac{14 - 3 \times 2}{2} = \frac{14 - 6}{2} = 4
     $$

6. **Calculate $27x^3 + 8y^3$ for both pairs $(x, y)$:**
   - For $(x, y) = \left(\frac{8}{3}, 3\right)$:
     $$
     27x^3 = 27 \left(\frac{8}{3}\right)^3 = 27 \times \frac{512}{27} = 512
     $$
     $$
     8y^3 = 8 \times 3^3 = 8 \times 27 = 216
     $$
     $$
     27x^3 + 8y^3 = 512 + 216 = 728
     $$

- For $(x, y) = (2, 4)$:
     $$
     27x^3 = 27 \times 2^3 = 27 \times 8 = 216
     $$
     $$
     8y^3 = 8 \times 4^3 = 8 \times 64 = 512
     $$
     $$
     27x^3 + 8y^3 = 216 + 512 = 728
     $$

### Conclusion:
Both pairs of $(x, y)$ satisfy the given equations and result in $27x^3 + 8y^3 = 728$. Therefore, Option 2 is indeed the correct answer.

### Explanation of Incorrect Options:
- **Option 1 (816):** This value does not match the calculated result of 728.
- **Option 3 (832):** This value does not match the calculated result of 728.
- **Option 4 (1228):** This value does not match the calculated result of 728.

Thus, the correct answer is Option 2, which is 728.