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(x-1)^2-4

2 min read Jun 17, 2024
(x-1)^2-4

Factoring and Solving (x-1)^2 - 4

The expression (x-1)^2 - 4 is a quadratic expression that can be factored and solved for its roots. Let's break down the process:

Factoring the Expression

  1. Recognize the pattern: The expression is in the form of a² - b², which is a difference of squares pattern.
  2. Apply the difference of squares formula: a² - b² = (a + b)(a - b)
  3. Substitute: In our case, a = (x-1) and b = 2.
  4. Factor: (x-1)² - 4 = ((x-1) + 2)((x-1) - 2)
  5. Simplify: (x-1)² - 4 = (x+1)(x-3)

Solving for the Roots

To find the roots, we set the factored expression equal to zero and solve for x:

(x+1)(x-3) = 0

This gives us two possible solutions:

  • x + 1 = 0 => x = -1
  • x - 3 = 0 => x = 3

Summary

Therefore, the factored form of (x-1)² - 4 is (x+1)(x-3). The roots of the expression are x = -1 and x = 3.

This demonstrates how recognizing common algebraic patterns like the difference of squares can simplify factoring and solving quadratic expressions.

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