(x-2)2-18=0

2 min read Jun 17, 2024
(x-2)2-18=0

Solving the Quadratic Equation: (x-2)² - 18 = 0

This article will guide you through the process of solving the quadratic equation (x-2)² - 18 = 0.

Understanding the Equation

The equation is in the form of a quadratic equation: ax² + bx + c = 0.

  • a = 1 (the coefficient of x²)
  • b = -4 (the coefficient of x)
  • c = -14 (the constant term)

Solving the Equation

We can solve this equation using several methods:

1. Expanding and Using the Quadratic Formula

  • Expand the square: (x-2)² = x² - 4x + 4
  • Rewrite the equation: x² - 4x + 4 - 18 = 0
  • Simplify: x² - 4x - 14 = 0
  • Apply the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a x = (4 ± √((-4)² - 4 * 1 * -14)) / 2 * 1 x = (4 ± √(80)) / 2 x = (4 ± 4√5) / 2 x = 2 ± 2√5

Therefore, the solutions to the equation are x = 2 + 2√5 and x = 2 - 2√5.

2. Taking the Square Root

  • Isolate the squared term: (x-2)² = 18
  • Take the square root of both sides: x - 2 = ±√18
  • Simplify: x - 2 = ±3√2
  • Solve for x: x = 2 ± 3√2

This method leads to the same solutions as the previous method: x = 2 + 3√2 and x = 2 - 3√2.

Conclusion

The quadratic equation (x-2)² - 18 = 0 has two solutions: x = 2 + 2√5 and x = 2 - 2√5. These solutions can be found by expanding the equation and using the quadratic formula or by taking the square root of both sides.

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