%5E4-13(x-1%2Fx%2B1)%5E2%2B36%3D0.jpeg "(x-1/x+1)^4-13(x-1/x+1)^2+36=0")
Solving the Equation: (x-1/x+1)^4 - 13(x-1/x+1)^2 + 36 = 0
This equation might seem intimidating at first, but we can solve it using a clever substitution. Let's break down the process:
1. Substitution
Let y = (x-1/x+1). Substituting this into our equation gives us:
y⁴ - 13y² + 36 = 0
2. Factoring the Quadratic
Now we have a quadratic equation in terms of y². This equation can be factored:
(y² - 9)(y² - 4) = 0
This gives us two possible solutions:
- y² - 9 = 0
- y² - 4 = 0
3. Solving for y
Solving these equations for y, we get:
- y² = 9 => y = ±3
- y² = 4 => y = ±2
4. Back Substitution
Now we need to substitute back our original expression for y:
- Case 1: y = 3 (x-1/x+1) = 3
- Case 2: y = -3 (x-1/x+1) = -3
- Case 3: y = 2 (x-1/x+1) = 2
- Case 4: y = -2 (x-1/x+1) = -2
5. Solving for x
Now, we need to solve each of these equations for x:
Case 1: (x-1/x+1) = 3
- Multiply both sides by (x+1): x-1 = 3(x+1)
- Expand and solve for x: x-1 = 3x + 3 => -4 = 2x => x = -2
Case 2: (x-1/x+1) = -3
- Follow similar steps as Case 1 to get x = 1/2
Case 3: (x-1/x+1) = 2
- Follow similar steps as Case 1 to get x = -3
Case 4: (x-1/x+1) = -2
- Follow similar steps as Case 1 to get x = 1
6. Solution
Therefore, the solutions to the equation (x-1/x+1)⁴ - 13(x-1/x+1)² + 36 = 0 are:
x = -2, x = 1/2, x = -3, and x = 1