Find the value of `xx^3` -`y^3`, if `xx`-y = 4 and xy =12

Options

A. 198
B. 208
C. 218
D. 228

Answer

To solve the problem, we need to find the value of \( x^3 - y^3 \) given that \( x - y = 4 \) and \( xy = 12 \).

### Step-by-Step Solution:

1. **Identify the given equations:**
We have:
\[ x - y = 4 \quad \text{(1)} \]
\[ xy = 12 \quad \text{(2)} \]

2. **Express \( x \) in terms of \( y \):**
From equation (1), we can express \( x \) as:
\[ x = y + 4 \quad \text{(3)} \]

3. **Substitute \( x \) in equation (2):**
Substitute equation (3) into equation (2):
\[ (y + 4)y = 12 \]
Expanding this gives:
\[ y^2 + 4y - 12 = 0 \quad \text{(4)} \]

4. **Solve the quadratic equation (4):**
We can solve the quadratic equation \( y^2 + 4y - 12 = 0 \) using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = 4, c = -12 \):
\[ y = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot (-12)}}{2 \cdot 1} \]
\[ y = \frac{-4 \pm \sqrt{16 + 48}}{2} \]
\[ y = \frac{-4 \pm \sqrt{64}}{2} \]
\[ y = \frac{-4 \pm 8}{2} \]
This gives us two possible values for \( y \):
\[ y = \frac{4}{2} = 2 \quad \text{or} \quad y = \frac{-12}{2} = -6 \]

5. **Find the corresponding \( x \) values:**
For \( y = 2 \):
\[ x = 2 + 4 = 6 \]
For \( y = -6 \):
\[ x = -6 + 4 = -2 \]

6. **Calculate \( x^3 - y^3 \):**
We will calculate \( x^3 - y^3 \) using the values \( x = 6 \) and \( y = 2 \):
\[ x^3 - y^3 = 6^3 - 2^3 \]
\[ = 216 - 8 \]
\[ = 208 \]

### Final Answer:
The value of \( x^3 - y^3 \) is \( 208 \).