If `x^(4) + y^(4)=19 and x+y=1` find `x^(2)y^(2)-2xy`

Answer

To solve the problem, we need to find the value of \( x^2y^2 - 2xy \) given the equations \( x^4 + y^4 = 19 \) and \( x + y = 1 \).

### Step-by-step Solution:

1. **Use the identity for \( (x + y)^2 \)**:
\[ (x + y)^2 = x^2 + y^2 + 2xy \]
Since \( x + y = 1 \), we have:
\[ 1^2 = x^2 + y^2 + 2xy \]
This simplifies to:
\[ 1 = x^2 + y^2 + 2xy \quad \text{(Equation 1)} \]

2. **Rearrange Equation 1 to find \( x^2 + y^2 \)**:
\[ x^2 + y^2 = 1 - 2xy \quad \text{(Equation 2)} \]

3. **Use the identity for \( x^4 + y^4 \)**:
\[ x^4 + y^4 = (x^2 + y^2)^2 - 2(xy)^2 \]
We know \( x^4 + y^4 = 19 \), so:
\[ (x^2 + y^2)^2 - 2(xy)^2 = 19 \quad \text{(Equation 3)} \]

4. **Substitute Equation 2 into Equation 3**:
First, substitute \( x^2 + y^2 = 1 - 2xy \) into Equation 3:
\[ (1 - 2xy)^2 - 2(xy)^2 = 19 \]

5. **Expand the equation**:
\[ (1 - 4xy + 4(xy)^2) - 2(xy)^2 = 19 \]
This simplifies to:
\[ 1 - 4xy + 2(xy)^2 = 19 \]

6. **Rearrange the equation**:
\[ 2(xy)^2 - 4xy + 1 - 19 = 0 \]
\[ 2(xy)^2 - 4xy - 18 = 0 \]

7. **Divide the entire equation by 2**:
\[ (xy)^2 - 2xy - 9 = 0 \]

8. **Let \( z = xy \)**, then we have:
\[ z^2 - 2z - 9 = 0 \]

9. **Solve the quadratic equation using the quadratic formula**:
\[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -2, c = -9 \):
\[ z = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-9)}}{2 \cdot 1} \]
\[ z = \frac{2 \pm \sqrt{4 + 36}}{2} \]
\[ z = \frac{2 \pm \sqrt{40}}{2} \]
\[ z = \frac{2 \pm 2\sqrt{10}}{2} \]
\[ z = 1 \pm \sqrt{10} \]

10. **Now, we need to find \( x^2y^2 - 2xy \)**:
Recall that \( x^2y^2 = (xy)^2 \):
\[ x^2y^2 - 2xy = z^2 - 2z \]
Substitute \( z = 1 + \sqrt{10} \) or \( z = 1 - \sqrt{10} \):
- For \( z = 1 + \sqrt{10} \):
\[ (1 + \sqrt{10})^2 - 2(1 + \sqrt{10}) = 1 + 2\sqrt{10} + 10 - 2 - 2\sqrt{10} = 9 \]
- For \( z = 1 - \sqrt{10} \):
\[ (1 - \sqrt{10})^2 - 2(1 - \sqrt{10}) = 1 - 2\sqrt{10} + 10 - 2 + 2\sqrt{10} = 9 \]

Thus, in both cases, we find:
\[ x^2y^2 - 2xy = 9 \]

### Final Answer:
\[ \boxed{9} \]