If `2a+b=12` and `ab=15`, then find the value of `8a^(3)+b^(3)` .

Answer

To solve the problem, we need to find the value of \( 8a^3 + b^3 \) given the equations \( 2a + b = 12 \) and \( ab = 15 \).

### Step-by-Step Solution:

1. **Identify the given equations:**
\[ 2a + b = 12 \quad \text{(1)} \]
\[ ab = 15 \quad \text{(2)} \]

2. **Use the identity for cubes:**
We can use the identity:
\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \]
In our case, let \( x = 2a \) and \( y = b \). Thus,
\[ (2a)^3 + b^3 = (2a + b)((2a)^2 - (2a)b + b^2) \]

3. **Calculate \( (2a + b) \):**
From equation (1), we already know:
\[ 2a + b = 12 \]

4. **Calculate \( (2a)^2 \), \( (2a)b \), and \( b^2 \):**
- \( (2a)^2 = 4a^2 \)
- \( (2a)b = 2ab = 2 \times 15 = 30 \) (using equation (2))
- \( b^2 \) can be expressed in terms of \( a \) and \( ab \). We will find \( b \) later.

5. **Substituting into the identity:**
Now we substitute into the identity:
\[ 8a^3 + b^3 = (2a + b)(4a^2 - 30 + b^2) \]

6. **Find \( b^2 \):**
We can express \( b \) from equation (1):
\[ b = 12 - 2a \]
Now substitute \( b \) into \( ab = 15 \):
\[ a(12 - 2a) = 15 \]
\[ 12a - 2a^2 = 15 \]
Rearranging gives:
\[ 2a^2 - 12a + 15 = 0 \]

7. **Solve the quadratic equation:**
We can use the quadratic formula \( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ a = \frac{12 \pm \sqrt{(-12)^2 - 4 \cdot 2 \cdot 15}}{2 \cdot 2} \]
\[ = \frac{12 \pm \sqrt{144 - 120}}{4} \]
\[ = \frac{12 \pm \sqrt{24}}{4} \]
\[ = \frac{12 \pm 2\sqrt{6}}{4} \]
\[ = 3 \pm \frac{\sqrt{6}}{2} \]

8. **Find the corresponding \( b \) values:**
Using \( b = 12 - 2a \):
- If \( a = 3 + \frac{\sqrt{6}}{2} \):
\[ b = 12 - 2\left(3 + \frac{\sqrt{6}}{2}\right) = 6 - \sqrt{6} \]
- If \( a = 3 - \frac{\sqrt{6}}{2} \):
\[ b = 12 - 2\left(3 - \frac{\sqrt{6}}{2}\right) = 6 + \sqrt{6} \]

9. **Calculate \( 8a^3 + b^3 \):**
Now substitute back into the identity:
\[ 8a^3 + b^3 = 12(4a^2 - 30 + b^2) \]
We need to calculate \( 4a^2 \) and \( b^2 \):
- \( 4a^2 = 4(3 \pm \frac{\sqrt{6}}{2})^2 \)
- \( b^2 = (6 \mp \sqrt{6})^2 \)

10. **Final calculation:**
After calculating \( 4a^2 \) and \( b^2 \), substitute back into the equation to find the final value of \( 8a^3 + b^3 \).

### Final Result:
After calculating, we find:
\[ 8a^3 + b^3 = 648 \]