(12-x)^2=8

3 min read Jun 16, 2024

Solving the Equation (12-x)^2 = 8

This equation involves a squared term, which we need to address before we can solve for x. Let's break down the steps:

First, we expand the left side of the equation using the formula (a-b)^2 = a^2 - 2ab + b^2:

(12-x)^2 = 12^2 - 2(12)(x) + x^2 = 144 - 24x + x^2

Now our equation becomes: 144 - 24x + x^2 = 8

Next, we want to set the equation to zero by moving all the terms to one side:

x^2 - 24x + 144 - 8 = 0

Simplifying: x^2 - 24x + 136 = 0

We have now arrived at a quadratic equation. There are a couple of ways to solve this:

Factoring: Try to find two numbers that add up to -24 and multiply to 136. In this case, it's not easy to find these numbers by inspection.
Quadratic Formula: The quadratic formula is a reliable way to solve for x in any quadratic equation of the form ax^2 + bx + c = 0:

x = [-b ± √(b^2 - 4ac)] / 2a

In our equation, a = 1, b = -24, and c = 136. Substituting these values into the quadratic formula:

x = [24 ± √((-24)^2 - 4 * 1 * 136)] / 2 * 1

x = [24 ± √(576 - 544)] / 2

x = [24 ± √32] / 2

x = [24 ± 4√2] / 2

x = 12 ± 2√2

Therefore, the solutions to the equation (12-x)^2 = 8 are:

x = 12 + 2√2 and x = 12 - 2√2