If ` a - b = 3 and a^(3) - b ^(3) = 117` then `| a + b|` is equal to

Options

A. 3
B. 5
C. 7
D. 9

Answer

To solve the problem step by step, we start with the given equations:

1. **Given equations:**
\[ a - b = 3 \quad \text{(Equation 1)} \]
\[ a^3 - b^3 = 117 \quad \text{(Equation 2)} \]

2. **Using the identity for the difference of cubes:**
We know that \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \). Substituting Equation 1 into this identity, we get:
\[ a^3 - b^3 = (3)(a^2 + ab + b^2) = 117 \]

3. **Dividing both sides by 3:**
\[ a^2 + ab + b^2 = \frac{117}{3} = 39 \quad \text{(Equation 3)} \]

4. **Substituting \( b \) from Equation 1:**
From Equation 1, we can express \( b \) in terms of \( a \):
\[ b = a - 3 \]
Now, substitute \( b \) into Equation 3:
\[ a^2 + a(a - 3) + (a - 3)^2 = 39 \]

5. **Expanding the equation:**
\[ a^2 + a^2 - 3a + (a^2 - 6a + 9) = 39 \]
Combine like terms:
\[ 3a^2 - 9a + 9 = 39 \]

6. **Rearranging the equation:**
\[ 3a^2 - 9a + 9 - 39 = 0 \]
\[ 3a^2 - 9a - 30 = 0 \]

7. **Dividing the entire equation by 3:**
\[ a^2 - 3a - 10 = 0 \]

8. **Factoring the quadratic equation:**
\[ (a - 5)(a + 2) = 0 \]
Thus, we have:
\[ a - 5 = 0 \quad \text{or} \quad a + 2 = 0 \]
Therefore, the solutions for \( a \) are:
\[ a = 5 \quad \text{or} \quad a = -2 \]

9. **Finding corresponding values of \( b \):**
- If \( a = 5 \):
\[ b = 5 - 3 = 2 \]
- If \( a = -2 \):
\[ b = -2 - 3 = -5 \]

10. **Calculating \( |a + b| \):**
- For \( a = 5 \) and \( b = 2 \):
\[ a + b = 5 + 2 = 7 \quad \Rightarrow \quad |a + b| = 7 \]
- For \( a = -2 \) and \( b = -5 \):
\[ a + b = -2 - 5 = -7 \quad \Rightarrow \quad |a + b| = 7 \]

Thus, in both cases, we find that:
\[ |a + b| = 7 \]

### Final Answer:
\[ |a + b| = 7 \]