Given `a - bb b =2 , a ^(3) - b ^(3) = 26` then `(a + b ) ^(2)` is

Options

A. 9
B. 4
C. 16
D. 12

Answer

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

1. \( a - b = 2 \)
2. \( a^3 - b^3 = 26 \)

We need to find \( (a + b)^2 \).

### Step 1: Use the identity for the difference of cubes

We know that:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]

### Step 2: Substitute \( a - b \) into the equation

From the first equation, we have \( a - b = 2 \). Substituting this into the identity gives:
\[ a^3 - b^3 = 2(a^2 + ab + b^2) \]

### Step 3: Set up the equation with the second given value

We know from the problem statement that \( a^3 - b^3 = 26 \). Therefore, we can set up the equation:
\[ 26 = 2(a^2 + ab + b^2) \]

### Step 4: Solve for \( a^2 + ab + b^2 \)

Dividing both sides of the equation by 2:
\[ a^2 + ab + b^2 = 13 \]

### Step 5: Use the square of a sum identity

We want to find \( (a + b)^2 \), which can be expressed as:
\[ (a + b)^2 = a^2 + 2ab + b^2 \]

### Step 6: Relate \( a^2 + b^2 \) to \( a^2 + ab + b^2 \)

We can express \( a^2 + b^2 \) in terms of \( a^2 + ab + b^2 \):
\[ a^2 + b^2 = (a^2 + ab + b^2) - ab \]
Substituting the value we found:
\[ a^2 + b^2 = 13 - ab \]

### Step 7: Substitute into the square of a sum formula

Now substituting back into the equation for \( (a + b)^2 \):
\[ (a + b)^2 = (13 - ab) + 2ab \]
\[ (a + b)^2 = 13 + ab \]

### Step 8: Find \( ab \)

We can find \( ab \) using the earlier derived equation from \( a^3 - b^3 \):
\[ 26 - 2 \cdot 3ab = 0 \quad \text{(from the rearrangement of the previous steps)} \]
This simplifies to:
\[ 18 = 6ab \implies ab = 3 \]

### Step 9: Substitute \( ab \) back into the equation for \( (a + b)^2 \)

Now substituting \( ab = 3 \) into the equation for \( (a + b)^2 \):
\[ (a + b)^2 = 13 + 3 = 16 \]

### Final Answer

Thus, the value of \( (a + b)^2 \) is:
\[ \boxed{16} \]