If X+Y=3,XY=2, Find the value of `X^(3)-Y^(3)`

Options

A. 7
B. 3
C. 5
D. 9

Answer

To solve the problem, we need to find the value of \( X^3 - Y^3 \) given that \( X + Y = 3 \) and \( XY = 2 \).

### Step 1: Use the identity for the difference of cubes
The formula for the difference of cubes is:
\[ X^3 - Y^3 = (X - Y)(X^2 + XY + Y^2) \]
We need to find \( X - Y \) and \( X^2 + XY + Y^2 \).

### Step 2: Find \( X - Y \)
We can use the identity:
\[ (X + Y)^2 = X^2 + 2XY + Y^2 \]
Substituting the known values:
\[ 3^2 = X^2 + 2(2) + Y^2 \]
This simplifies to:
\[ 9 = X^2 + 4 + Y^2 \]
Thus,
\[ X^2 + Y^2 = 9 - 4 = 5 \]

Now, we can find \( X^2 + XY + Y^2 \):
\[ X^2 + XY + Y^2 = X^2 + Y^2 + XY = 5 + 2 = 7 \]

### Step 3: Find \( X - Y \) using \( (X + Y)^2 - 4XY \)
We can also find \( X - Y \) using:
\[ (X - Y)^2 = (X + Y)^2 - 4XY \]
Substituting the known values:
\[ (X - Y)^2 = 3^2 - 4(2) = 9 - 8 = 1 \]
Taking the square root gives:
\[ X - Y = 1 \quad \text{or} \quad X - Y = -1 \]

### Step 4: Substitute back into the difference of cubes formula
Now we can substitute \( X - Y \) and \( X^2 + XY + Y^2 \) into the difference of cubes 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
\[ X^3 - Y^3 = 7 \]