(x-1)^4-8(x-1)^2-9=0

2 min read Jun 17, 2024

Solving the Equation (x-1)^4 - 8(x-1)^2 - 9 = 0

This equation might look intimidating at first, but it can be solved using a simple substitution. Let's break it down step-by-step:

We can simplify the equation by substituting a new variable. Let's substitute:

y = (x-1)^2

Now, our equation becomes:

y^2 - 8y - 9 = 0

This is now a quadratic equation, which is much easier to solve.

We can solve this quadratic equation using the quadratic formula:

y = (-b ± √(b^2 - 4ac)) / 2a

Where a = 1, b = -8, and c = -9.

Plugging these values into the formula, we get:

y = (8 ± √((-8)^2 - 4 * 1 * -9)) / 2 * 1

y = (8 ± √(100)) / 2

y = (8 ± 10) / 2

This gives us two possible solutions for y:

Now we need to substitute back the original variable (x-1)^2 for y and solve for x:

For y1 = 9:
- (x-1)^2 = 9
- x-1 = ±3
- x1 = 4
- x2 = -2
For y2 = -1:
- (x-1)^2 = -1
- This equation has no real solutions, as the square of a real number cannot be negative.

Therefore, the solutions to the equation (x-1)^4 - 8(x-1)^2 - 9 = 0 are x = 4 and x = -2.