(x-1)2 = 4√(x-4) 2 Simplify

3 min read Jun 17, 2024

Solving the Equation: (x-1)² = 4√(x-4)

This problem involves a square root and a squared term, so we'll need to employ some algebraic techniques to solve it. Here's how we can approach it:

1. Isolate the Square Root

First, we want to get the square root term by itself on one side of the equation.

Divide both sides by 4:
(x-1)² / 4 = √(x-4)

2. Square Both Sides

To eliminate the square root, we square both sides of the equation.

Square both sides:
((x-1)² / 4)² = (√(x-4))²
Simplify: (x-1)⁴ / 16 = x - 4

3. Rearrange and Simplify

Now we have a polynomial equation. Let's rearrange it to make it easier to solve.

Multiply both sides by 16: (x-1)⁴ = 16(x - 4)
Expand the left side: x⁴ - 4x³ + 6x² - 4x + 1 = 16x - 64
Move all terms to one side: x⁴ - 4x³ + 6x² - 20x + 65 = 0

4. Finding Solutions

Unfortunately, this is a fourth-degree polynomial equation, and there's no simple formula to find its solutions directly. We'll need to use numerical methods or graphing techniques to approximate the solutions.

Here are some common approaches:

Graphing: Plot the function y = x⁴ - 4x³ + 6x² - 20x + 65. The x-intercepts of the graph represent the solutions to the equation.
Numerical Methods: Use techniques like the Newton-Raphson method or a numerical solver to find approximate solutions.

Important Note:

Always check your solutions by substituting them back into the original equation to ensure they are valid. You may find that some solutions are extraneous, meaning they don't satisfy the original equation.