(x^2+1/x^2)-3(x-1/x)-2=0

3 min read Jun 17, 2024

Solving the Equation: (x^2 + 1/x^2) - 3(x - 1/x) - 2 = 0

This equation might look intimidating at first glance, but we can solve it by using a clever substitution and applying quadratic formula. Here's how:

1. Substitution

Let's make the following substitution:

y = x - 1/x

Now, let's square this equation:

y^2 = (x - 1/x)^2 = x^2 - 2 + 1/x^2

Notice that we can rewrite the original equation in terms of 'y':

(x^2 + 1/x^2) - 3(x - 1/x) - 2 = 0

y^2 - 2 - 3y - 2 = 0

y^2 - 3y - 4 = 0

2. Solving the Quadratic Equation

We now have a simple quadratic equation in terms of 'y'. We can solve this using the quadratic formula:

y = (-b ± √(b^2 - 4ac)) / 2a

Where a = 1, b = -3, and c = -4.

Plugging in these values, we get:

y = (3 ± √((-3)^2 - 4 * 1 * -4)) / (2 * 1)

y = (3 ± √25) / 2

y = (3 ± 5) / 2

This gives us two possible solutions for 'y':

y1 = 4
y2 = -1

3. Back-Substitution

Now, we need to substitute back our original expression for 'y':

y1 = x - 1/x = 4

y2 = x - 1/x = -1

Solving these two equations for 'x' will give us the solutions for the original equation.

For y1 = 4:

x - 1/x = 4
x^2 - 4x - 1 = 0

We can use the quadratic formula again to solve for 'x' in this equation.

For y2 = -1:

x - 1/x = -1
x^2 + x - 1 = 0

Again, we can use the quadratic formula to solve for 'x' in this equation.

Conclusion

By using substitution and solving the resulting quadratic equations, we can find the solutions for the original equation. Remember to solve for 'x' from both values of 'y' to get all possible solutions.