(x-1)^2-(x+2)^2=6

2 min read Jun 17, 2024

Solving the Equation (x-1)^2 - (x+2)^2 = 6

This article will guide you through solving the equation (x-1)^2 - (x+2)^2 = 6.

Understanding the Equation

The equation involves squaring binomials and subtracting them. This presents a good opportunity to utilize the difference of squares factorization pattern.

Applying the Difference of Squares Pattern

The difference of squares pattern states: a² - b² = (a+b)(a-b).

Let's apply this pattern to our equation:

Identify 'a' and 'b':
- a = (x-1)
- b = (x+2)
Substitute into the pattern:
- (x-1)² - (x+2)² = [(x-1) + (x+2)][(x-1) - (x+2)]
Simplify the expression:
- [(x-1) + (x+2)][(x-1) - (x+2)] = (2x + 1)(-3)
Expand the simplified expression:
- (2x + 1)(-3) = -6x - 3

Solving the Equation

Now our equation is simplified to -6x - 3 = 6. Let's solve for x:

Add 3 to both sides:
- -6x = 9
Divide both sides by -6:
- x = -3/2

Solution

Therefore, the solution to the equation (x-1)² - (x+2)² = 6 is x = -3/2.

Verification

We can verify our solution by substituting x = -3/2 back into the original equation:

[( -3/2 - 1 )² - ( -3/2 + 2 )²] = [(-5/2)² - (1/2)²] = [25/4 - 1/4] = 6

This confirms that our solution is correct.