(x2+1/x2)-5(x+1/x)+6=0

3 min read Jun 17, 2024

Solving the Equation: (x² + 1/x²) - 5(x + 1/x) + 6 = 0

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

1. Substitution:

Let's make a substitution to simplify the equation. We can define:

y = x + 1/x

Now, let's square both sides of this equation:

y² = (x + 1/x)² = x² + 2(x)(1/x) + 1/x² = x² + 2 + 1/x²

Notice that we can rearrange this to get:

x² + 1/x² = y² - 2

2. Rewriting the Equation:

Substituting these expressions back into our original equation, we get:

(y² - 2) - 5y + 6 = 0

This simplifies to a quadratic equation:

y² - 5y + 4 = 0

3. Solving the Quadratic:

Now, we can easily factor this quadratic:

(y - 4)(y - 1) = 0

This gives us two possible solutions for y:

y = 4
y = 1

4. Finding x:

Now, we need to find the values of x that correspond to these values of y. Let's remember our substitution:

y = x + 1/x

For y = 4:

4 = x + 1/x

Multiplying both sides by x:

4x = x² + 1

Rearranging into a quadratic:

x² - 4x + 1 = 0

Using the quadratic formula, we get:

x = (4 ± √12)/2 = 2 ± √3

For y = 1:

1 = x + 1/x

Following a similar procedure as above, we get:

x² - x + 1 = 0

Again, using the quadratic formula, we get:

x = (1 ± √-3)/2 = (1 ± i√3)/2

where 'i' represents the imaginary unit (√-1).

Conclusion:

Therefore, the solutions to the equation (x² + 1/x²) - 5(x + 1/x) + 6 = 0 are:

x = 2 + √3
x = 2 - √3
x = (1 + i√3)/2
x = (1 - i√3)/2