(x/x+1)^2-5(x/x+1)+6=0

2 min read Jun 17, 2024

Solving the Equation: (x/(x+1))^2 - 5(x/(x+1)) + 6 = 0

This equation may look intimidating at first, but it can be solved using a simple substitution technique. Let's break down the steps:

1. Substitution:

Let y = x/(x+1). This simplifies the equation to:

y² - 5y + 6 = 0

2. Factoring the Quadratic:

This is now a standard quadratic equation. We can factor it as:

(y - 2)(y - 3) = 0

This gives us two possible solutions for y:

y = 2
y = 3

3. Back-Substituting to Find x:

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

For y = 2:
- 2 = x/(x+1)
- 2x + 2 = x
- x = -2
For y = 3:
- 3 = x/(x+1)
- 3x + 3 = x
- 2x = -3
- x = -3/2

4. Checking for Extraneous Solutions:

It's crucial to check our solutions in the original equation to ensure they are valid. We find that both solutions, x = -2 and x = -3/2, satisfy the original equation.

Conclusion:

Therefore, the solutions to the equation (x/(x+1))^2 - 5(x/(x+1)) + 6 = 0 are x = -2 and x = -3/2.