If x+y=3, xy=2, find the value of `x^3-y^3`.

Options

A. 13
B. 11
C. 5
D. 7

Answer

To find the value of \( x^3 - y^3 \) given that \( x + y = 3 \) and \( xy = 2 \), we can use the formula for the difference of cubes:

\[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) \]

### Step 1: Find \( x - y \)

We know:
\[ x + y = 3 \]
\[ xy = 2 \]

Using the identity \( (x + y)^2 = x^2 + 2xy + y^2 \), we can express \( x^2 + y^2 \):

\[ (x + y)^2 = x^2 + 2xy + y^2 \]
\[ 3^2 = x^2 + 2(2) + y^2 \]
\[ 9 = x^2 + 4 + y^2 \]
\[ x^2 + y^2 = 9 - 4 = 5 \]

Now, we can find \( x^2 + xy + y^2 \):

\[ x^2 + xy + y^2 = x^2 + y^2 + xy = 5 + 2 = 7 \]

### Step 2: Find \( x - y \)

To find \( x - y \), we can use the identity \( (x - y)^2 = (x + y)^2 - 4xy \):

\[ (x - y)^2 = (x + y)^2 - 4xy \]
\[ = 3^2 - 4(2) \]
\[ = 9 - 8 = 1 \]
\[ x - y = \sqrt{1} = 1 \quad \text{or} \quad x - y = -\sqrt{1} = -1 \]

### Step 3: Calculate \( x^3 - y^3 \)

Now we can substitute \( x - y \) and \( x^2 + xy + y^2 \) into the formula:

\[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) = (1)(7) = 7 \]

Thus, the value of \( x^3 - y^3 \) is \( 7 \).

### Final Answer:
\[ \boxed{7} \]