(2x-1)^2-81=0

2 min read Jun 16, 2024

Solving the Quadratic Equation: (2x-1)^2 - 81 = 0

This article will guide you through solving the quadratic equation (2x-1)^2 - 81 = 0.

Understanding the Equation

The equation is a quadratic equation, meaning it has a highest power of 2 in its variable (x). It's also in a special form, where the left side is a perfect square trinomial minus a constant. This allows us to solve it using a simple method.

Solving the Equation

Isolate the perfect square: Add 81 to both sides of the equation: (2x-1)^2 = 81
Take the square root of both sides: Remember to include both positive and negative roots: 2x-1 = ±9
Solve for x:
- Case 1: 2x-1 = 9 Adding 1 to both sides: 2x = 10 Dividing both sides by 2: x = 5
- Case 2: 2x-1 = -9 Adding 1 to both sides: 2x = -8 Dividing both sides by 2: x = -4

Solutions

Therefore, the solutions to the quadratic equation (2x-1)^2 - 81 = 0 are x = 5 and x = -4.

Verification

You can verify these solutions by substituting them back into the original equation:

For x = 5: (2(5)-1)^2 - 81 = 9^2 - 81 = 81 - 81 = 0
For x = -4: (2(-4)-1)^2 - 81 = (-9)^2 - 81 = 81 - 81 = 0

Since both solutions result in 0, we have confirmed their validity.