If `x^(4)+x^(2)y^(2)+y^(4)=133` and `x^(2)-xy+y^(2)=7,` then what is the value of xy ?

Options

A. 8
B. 12
C. 4
D. 6

Answer

To solve the problem, we start with the two given equations:

1. \( x^4 + x^2y^2 + y^4 = 133 \)
2. \( x^2 - xy + y^2 = 7 \)

We need to find the value of \( xy \).

### Step 1: Rewrite the first equation
We can use the identity:
\[ x^4 + y^4 + x^2y^2 = (x^2 + y^2)^2 - x^2y^2 \]
This means we can express \( x^4 + x^2y^2 + y^4 \) in terms of \( x^2 + y^2 \) and \( xy \).

### Step 2: Express \( x^2 + y^2 \)
From the second equation \( x^2 - xy + y^2 = 7 \), we can express \( x^2 + y^2 \) as:
\[ x^2 + y^2 = 7 + xy \]

### Step 3: Substitute into the first equation
Now substitute \( x^2 + y^2 \) into the rewritten first equation:
\[ (x^2 + y^2)^2 - x^2y^2 = 133 \]
Substituting \( x^2 + y^2 = 7 + xy \):
\[ (7 + xy)^2 - x^2y^2 = 133 \]

### Step 4: Expand and simplify
Expanding \( (7 + xy)^2 \):
\[ 49 + 14xy + (xy)^2 - x^2y^2 = 133 \]
We know that \( x^2y^2 = (xy)^2 \), so we can rewrite it as:
\[ 49 + 14xy + (xy)^2 - (xy)^2 = 133 \]
This simplifies to:
\[ 49 + 14xy = 133 \]

### Step 5: Solve for \( xy \)
Now, isolate \( xy \):
\[ 14xy = 133 - 49 \]
\[ 14xy = 84 \]
\[ xy = \frac{84}{14} = 6 \]

Thus, the value of \( xy \) is \( 6 \).

### Final Answer
The value of \( xy \) is \( 6 \).

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