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

Options

A. 16 ab
B. 0
C. 1
D. 2

Answer

To solve the equation \( a = \frac{b^2}{b - a} \) and find the value of \( a^3 + b^3 \), we can follow these steps:

### Step 1: Rearranging the equation
Starting with the equation:
\[ a = \frac{b^2}{b - a} \]
We can multiply both sides by \( b - a \) to eliminate the fraction:
\[ a(b - a) = b^2 \]
Expanding the left side gives us:
\[ ab - a^2 = b^2 \]

### Step 2: Rearranging to form a quadratic equation
Rearranging the equation, we have:
\[ a^2 - ab + b^2 = 0 \]

### Step 3: Recognizing the quadratic form
This is a quadratic equation in terms of \( a \). The standard form of a quadratic equation is \( Ax^2 + Bx + C = 0 \), where:
- \( A = 1 \)
- \( B = -b \)
- \( C = b^2 \)

### Step 4: Using the identity for \( a^3 + b^3 \)
We know that:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
From our earlier step, we have \( a^2 - ab + b^2 = 0 \).

### Step 5: Substituting into the identity
Substituting \( a^2 - ab + b^2 = 0 \) into the identity gives us:
\[ a^3 + b^3 = (a + b)(0) = 0 \]

### Conclusion
Thus, the value of \( a^3 + b^3 \) is:
\[ \boxed{0} \]

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