(2x-1)^2=(x-1)^2+8

3 min read Jun 16, 2024

Solving the Equation: (2x-1)^2 = (x-1)^2 + 8

This article will guide you through the steps of solving the equation (2x-1)^2 = (x-1)^2 + 8. We will use algebraic manipulation to find the values of x that satisfy this equation.

Expanding the Equation

First, we need to expand the squares on both sides of the equation.

Left-hand side: (2x-1)^2 = (2x-1)(2x-1) = 4x^2 - 4x + 1
Right-hand side: (x-1)^2 = (x-1)(x-1) = x^2 - 2x + 1

Now our equation looks like this: 4x^2 - 4x + 1 = x^2 - 2x + 1 + 8

Simplifying the Equation

Let's combine like terms to simplify the equation:

Subtract x^2 from both sides: 3x^2 - 4x + 1 = -2x + 1 + 8
Add 2x to both sides: 3x^2 - 2x + 1 = 1 + 8
Subtract 1 from both sides: 3x^2 - 2x = 8

Solving the Quadratic Equation

We now have a quadratic equation in the form of ax^2 + bx + c = 0. There are several ways to solve this:

Factoring: In this case, factoring might be a bit tricky.
Quadratic Formula: This is a reliable method for solving any quadratic equation. The formula is:
- x = (-b ± √(b^2 - 4ac)) / 2a
- Where a = 3, b = -2, and c = -8

Let's use the quadratic formula to solve for x:

Substitute the values: x = (2 ± √((-2)^2 - 4 * 3 * -8)) / (2 * 3)
Simplify: x = (2 ± √(100)) / 6
Calculate: x = (2 ± 10) / 6
Solve for both possible solutions:
- x1 = (2 + 10) / 6 = 2
- x2 = (2 - 10) / 6 = -4/3

Solution

Therefore, the solutions to the equation (2x-1)^2 = (x-1)^2 + 8 are x = 2 and x = -4/3.