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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.