(x-5)^4-3(x-5)^2-4=0

2 min read Jun 17, 2024

Solving the Equation (x-5)^4 - 3(x-5)^2 - 4 = 0

This equation might look intimidating at first, but we can solve it using a clever substitution.

Notice that the equation contains terms with (x-5) raised to even powers. Let's make a substitution to simplify things:

Let **y = (x-5)**²

Now we can rewrite the equation:

y² - 3y - 4 = 0

This is a simple quadratic equation that we can easily solve.

We can factor the quadratic equation as follows:

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

This gives us two possible solutions for y:

Now, we need to substitute back to find the solutions for x:

For y = 4:
- (x-5)² = 4
- x-5 = ±2
- x = 7 or x = 3
For y = -1:
- (x-5)² = -1
- This equation has no real solutions, as the square of a real number cannot be negative.

Therefore, the solutions to the original equation (x-5)^4 - 3(x-5)² - 4 = 0 are x = 7 and x = 3.