(x+3)^2-36=0

2 min read Jun 16, 2024
(x+3)^2-36=0

Solving the Equation (x+3)^2 - 36 = 0

This equation is a quadratic equation in disguise. Let's break down how to solve it:

1. Simplifying the Equation

  • Expand the square: (x+3)^2 = (x+3)(x+3) = x^2 + 6x + 9
  • Substitute: Now the equation becomes: x^2 + 6x + 9 - 36 = 0
  • Combine like terms: x^2 + 6x - 27 = 0

2. Solving the Quadratic Equation

We have a standard quadratic equation now: ax^2 + bx + c = 0, where a = 1, b = 6, and c = -27. We can solve this using various methods:

  • Factoring:

    • Find two numbers that multiply to -27 and add up to 6. These numbers are 9 and -3.
    • Factor the equation: (x + 9)(x - 3) = 0
    • Set each factor to zero and solve:
      • x + 9 = 0 => x = -9
      • x - 3 = 0 => x = 3
  • Quadratic Formula:

    • The quadratic formula solves for x in any equation of the form ax^2 + bx + c = 0:
      • x = (-b ± √(b^2 - 4ac)) / 2a
    • Substitute the values:
      • x = (-6 ± √(6^2 - 4 * 1 * -27)) / 2 * 1
      • x = (-6 ± √(144)) / 2
      • x = (-6 ± 12) / 2
      • x = 3 or x = -9

3. The Solutions

Therefore, the solutions to the equation (x+3)^2 - 36 = 0 are x = 3 and x = -9.

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