If `a^(3) + b^(3) = 5824` and `a+b = 28` then `(a-b)^(2) + ab` is equal to :

Options

A. 208
B. 152
C. 180
D. 236

Answer

To solve the problem, we need to find the value of \((a-b)^2 + ab\) given that \(a^3 + b^3 = 5824\) and \(a + b = 28\).

### Step 1: Use the identity for the sum of cubes
We know that:
\[ a^3 + b^3 = (a+b)(a^2 - ab + b^2) \]
Substituting the known values:
\[ 5824 = 28(a^2 - ab + b^2) \]

### Step 2: Simplify the equation
Now, we can simplify the equation:
\[ a^2 - ab + b^2 = \frac{5824}{28} \]
Calculating the right side:
\[ \frac{5824}{28} = 208 \]
So we have:
\[ a^2 - ab + b^2 = 208 \]

### Step 3: Use the square of the sum
We also know that:
\[ a^2 + b^2 = (a+b)^2 - 2ab \]
Substituting \(a + b = 28\):
\[ a^2 + b^2 = 28^2 - 2ab = 784 - 2ab \]

### Step 4: Substitute into the equation
Now we can substitute \(a^2 + b^2\) into the equation from Step 2:
\[ 784 - 2ab - ab = 208 \]
This simplifies to:
\[ 784 - 3ab = 208 \]

### Step 5: Solve for \(ab\)
Rearranging gives:
\[ 3ab = 784 - 208 \]
Calculating the right side:
\[ 3ab = 576 \]
Thus:
\[ ab = \frac{576}{3} = 192 \]

### Step 6: Find \((a-b)^2 + ab\)
Now we need to find \((a-b)^2 + ab\). We can use the identity:
\[ (a-b)^2 = (a+b)^2 - 4ab \]
Substituting the known values:
\[ (a-b)^2 = 28^2 - 4 \cdot 192 \]
Calculating:
\[ (a-b)^2 = 784 - 768 = 16 \]

### Step 7: Calculate \((a-b)^2 + ab\)
Now we can find:
\[ (a-b)^2 + ab = 16 + 192 = 208 \]

Thus, the final answer is:
\[ \boxed{208} \]