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

2 min read Jun 17, 2024
(x-5)^4-3(x-5)^2-4=0

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.

The Substitution Trick

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.

Solving the Quadratic Equation

We can factor the quadratic equation as follows:

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

This gives us two possible solutions for y:

  • y = 4
  • y = -1

Substituting Back

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.

The Solutions

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

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