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(x−6)2−5=0

2 min read Jun 17, 2024
(x−6)2−5=0

Solving the Quadratic Equation (x−6)²−5=0

This article will guide you through solving the quadratic equation (x−6)²−5=0. We will use the following steps:

1. Simplify the Equation

  • Expand the square: (x−6)² = (x−6)(x−6) = x² - 12x + 36
  • Substitute: The equation becomes: x² - 12x + 36 - 5 = 0
  • Combine like terms: x² - 12x + 31 = 0

2. Solve using the Quadratic Formula

Now that the equation is in standard quadratic form (ax² + bx + c = 0), we can use the quadratic formula to find the solutions for x:

  • Quadratic Formula: x = (-b ± √(b² - 4ac)) / 2a
  • Identify a, b, and c:
    • a = 1
    • b = -12
    • c = 31
  • Substitute the values into the formula:
    • x = (12 ± √((-12)² - 4 * 1 * 31)) / (2 * 1)
    • x = (12 ± √(144 - 124)) / 2
    • x = (12 ± √20) / 2
    • x = (12 ± 2√5) / 2
  • Simplify:
    • x = 6 ± √5

3. Solutions

Therefore, the solutions to the equation (x−6)²−5=0 are:

  • x = 6 + √5
  • x = 6 - √5

These are the two values of x that satisfy the original equation.

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