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(x-9)^2-49=0

2 min read Jun 17, 2024
(x-9)^2-49=0

Solving the Equation: (x-9)^2 - 49 = 0

This equation is a quadratic equation in disguise. We can solve it using a few different methods:

Method 1: Factoring

  1. Recognize the pattern: The equation resembles the difference of squares pattern: a² - b² = (a+b)(a-b)
  2. Apply the pattern: We can rewrite the equation as: [(x-9) + 7][(x-9) - 7] = 0
  3. Simplify: (x-2)(x-16) = 0
  4. Solve for x: This gives us two possible solutions: x = 2 or x = 16

Method 2: Using the Square Root Property

  1. Isolate the squared term: Add 49 to both sides of the equation: (x-9)² = 49
  2. Take the square root: Take the square root of both sides: x - 9 = ±7
  3. Solve for x: This gives us two possible solutions: x = 9 + 7 = 16 or x = 9 - 7 = 2

Method 3: Expanding and Solving the Quadratic Equation

  1. Expand the square: (x-9)² = x² - 18x + 81
  2. Rewrite the equation: x² - 18x + 81 - 49 = 0
  3. Simplify: x² - 18x + 32 = 0
  4. Factor the quadratic: (x-16)(x-2) = 0
  5. Solve for x: This gives us two possible solutions: x = 16 or x = 2

Therefore, the solutions to the equation (x-9)² - 49 = 0 are x = 2 and x = 16.

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