If x^2y^2 + 1/(x^2y^2) = 83, then the value of xy - 1/xy is?

Steps to Solve

1. Define a new variable for simplification

Let $z = xy$. Then, we can rewrite the given equation $x^2y^2 + \frac{1}{x^2y^2} = 83$ in terms of $z$: [ x^2y^2 = (xy)^2 = z^2 ] Thus, we can express the original equation as: [ z^2 + \frac{1}{z^2} = 83 ]

2. Multiply through by (z^2)

To eliminate the fraction, multiply both sides of the equation by $z^2$: [ z^4 + 1 = 83z^2 ]

3. Rearrange the equation to standard form

Rearranging gives: [ z^4 - 83z^2 + 1 = 0 ]

4. Substitute (u = z^2)

Let (u = z^2). Then the equation becomes: [ u^2 - 83u + 1 = 0 ]

5. Use the quadratic formula

Using the quadratic formula (u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}), where (a = 1, b = -83, c = 1): [ u = \frac{83 \pm \sqrt{(-83)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} ] Calculating the discriminant: [ = \frac{83 \pm \sqrt{6889 - 4}}{2} = \frac{83 \pm \sqrt{6885}}{2} ]

6. Find (z) from (u)

Since (u = z^2), we find: [ z = \sqrt{\frac{83 \pm \sqrt{6885}}{2}} ]

7. Find the value of (xy - \frac{1}{xy})

We want to find the value of: [ xy - \frac{1}{xy} = z - \frac{1}{z} ] This can also be expressed as: [ = \frac{z^2 - 1}{z} ] Substituting (z^2 = u): [ = \frac{u - 1}{\sqrt{u}} ]

After calculating both possible values, we can find the respective values for $z$.

8. Calculate (z - \frac{1}{z})

Using either value of (z): [ xy - \frac{1}{xy} = z - \frac{1}{z} = \sqrt{u} - \frac{1}{\sqrt{u}} ]