If `a ^(3) - b ^(3) = 56 and a - b = 2 ` then what is the value of `a ^(2) +b ^(2)`

Options

A. 12
B. 20
C. 28
D. 32

Answer

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

1. \( a^3 - b^3 = 56 \)
2. \( a - b = 2 \)

We want to find the value of \( a^2 + b^2 \).

### Step 1: Use the identity for the difference of cubes
We can use the identity for the difference of cubes:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Substituting the known values into this identity:
\[ 56 = (2)(a^2 + ab + b^2) \]

### Step 2: Simplify the equation
Now, we can simplify:
\[ 56 = 2(a^2 + ab + b^2) \]
Dividing both sides by 2:
\[ a^2 + ab + b^2 = 28 \quad \text{(Equation 1)} \]

### Step 3: Use the square of the difference
Next, we can use the square of the difference:
\[ (a - b)^2 = a^2 - 2ab + b^2 \]
Substituting \( a - b = 2 \):
\[ 2^2 = a^2 - 2ab + b^2 \]
This simplifies to:
\[ 4 = a^2 - 2ab + b^2 \quad \text{(Equation 2)} \]

### Step 4: Express \( a^2 + b^2 \)
From Equation 1, we can express \( a^2 + b^2 \):
\[ a^2 + b^2 = 28 - ab \]

### Step 5: Substitute into Equation 2
Now we can substitute \( a^2 + b^2 \) from Equation 1 into Equation 2:
\[ 4 = (28 - ab) - 2ab \]
This simplifies to:
\[ 4 = 28 - 3ab \]
Rearranging gives:
\[ 3ab = 28 - 4 \]
\[ 3ab = 24 \]
Dividing by 3:
\[ ab = 8 \quad \text{(Equation 3)} \]

### Step 6: Substitute \( ab \) back into Equation 1
Now we substitute \( ab = 8 \) back into Equation 1:
\[ a^2 + b^2 = 28 - 8 \]
\[ a^2 + b^2 = 20 \]

### Final Answer
Thus, the value of \( a^2 + b^2 \) is:
\[ \boxed{20} \]

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