If `x^(4) + x^(2) y^(2) + y^(4) = 91 and x^(2) - xy + y^(2) = 13`, then what is the value of `| x-y|?`

Options

A. 8
B. 6
C. 2
D. 4

Answer

To solve the equations given in the problem, we will follow these steps:

1. **Identify the equations**:
We have two equations:
- \( x^4 + x^2y^2 + y^4 = 91 \) (Equation 1)
- \( x^2 - xy + y^2 = 13 \) (Equation 2)

2. **Rewrite Equation 1**:
Notice that \( x^4 + x^2y^2 + y^4 \) can be factored using the identity:
\[ x^4 + y^4 + x^2y^2 = (x^2 + y^2)^2 - (xy)^2 \]
Therefore, we can rewrite Equation 1 as:
\[ (x^2 + y^2)^2 - (xy)^2 = 91 \]

3. **Rewrite Equation 2**:
We can also rewrite Equation 2 using another identity:
\[ x^2 + y^2 = (x^2 - xy + y^2) + xy = 13 + xy \]
Thus, we can express \( x^2 + y^2 \) in terms of \( xy \):
\[ x^2 + y^2 = 13 + xy \]

4. **Substituting into Equation 1**:
Substitute \( x^2 + y^2 \) from Equation 2 into the rewritten Equation 1:
\[ (13 + xy)^2 - (xy)^2 = 91 \]
Expanding this gives:
\[ 169 + 26xy + (xy)^2 - (xy)^2 = 91 \]
Simplifying this results in:
\[ 169 + 26xy = 91 \]
Rearranging gives:
\[ 26xy = 91 - 169 \]
\[ 26xy = -78 \]
\[ xy = -3 \]

5. **Finding \( x^2 + y^2 \)**:
Now substitute \( xy = -3 \) back into the expression for \( x^2 + y^2 \):
\[ x^2 + y^2 = 13 + (-3) = 10 \]

6. **Using the identities**:
We know:
- \( x^2 + y^2 = 10 \)
- \( xy = -3 \)

Now we can find \( x \) and \( y \) using the quadratic equation:
\[ t^2 - (x^2 + y^2)t + xy = 0 \]
This becomes:
\[ t^2 - 10t - 3 = 0 \]

7. **Solving the quadratic equation**:
Using the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ t = \frac{10 \pm \sqrt{10^2 - 4 \cdot 1 \cdot (-3)}}{2 \cdot 1} \]
\[ t = \frac{10 \pm \sqrt{100 + 12}}{2} \]
\[ t = \frac{10 \pm \sqrt{112}}{2} \]
\[ t = \frac{10 \pm 4\sqrt{7}}{2} \]
\[ t = 5 \pm 2\sqrt{7} \]

8. **Finding \( |x - y| \)**:
The roots \( x \) and \( y \) are \( 5 + 2\sqrt{7} \) and \( 5 - 2\sqrt{7} \). The difference is:
\[ |x - y| = |(5 + 2\sqrt{7}) - (5 - 2\sqrt{7})| = |4\sqrt{7}| \]

Thus, the final answer is:
\[ |x - y| = 4\sqrt{7} \]