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Solving the Quadratic Equation: (x-5)² - 49 = 0
This article will guide you through the steps of solving the quadratic equation (x-5)² - 49 = 0.
Understanding the Equation
The equation (x-5)² - 49 = 0 is a quadratic equation because the highest power of the variable x is 2. We can solve this equation using various methods, such as:
- Factoring: Recognizing the equation as a difference of squares.
- Square Root Property: Isolating the squared term and taking the square root of both sides.
- Quadratic Formula: A general formula to solve any quadratic equation.
Solving by Factoring
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Recognize the Difference of Squares: The equation can be rewritten as [(x-5) + 7][(x-5) - 7] = 0, which is a difference of squares pattern (a² - b² = (a+b)(a-b)).
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Set Each Factor to Zero: We now have two factors: (x-5) + 7 = 0 and (x-5) - 7 = 0.
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Solve for x:
- (x-5) + 7 = 0 => x = -2
- (x-5) - 7 = 0 => x = 12
Therefore, the solutions to the equation (x-5)² - 49 = 0 are x = -2 and x = 12.
Solving by Square Root Property
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Isolate the Squared Term: Add 49 to both sides of the equation to get (x-5)² = 49.
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Take the Square Root: Take the square root of both sides, remembering to consider both positive and negative roots: x-5 = ±7.
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Solve for x:
- x-5 = 7 => x = 12
- x-5 = -7 => x = -2
This again gives us the solutions x = -2 and x = 12.
Solving by Quadratic Formula
The quadratic formula provides a universal solution for equations in the form ax² + bx + c = 0. In our case, we have a = 1, b = -10, and c = -24 (after expanding the equation).
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Apply the Formula: x = (-b ± √(b² - 4ac)) / 2a x = (10 ± √((-10)² - 4 * 1 * -24)) / 2 * 1
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Simplify: x = (10 ± √(196)) / 2 x = (10 ± 14) / 2
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Solve for x: x = (10 + 14) / 2 = 12 x = (10 - 14) / 2 = -2
Once again, we obtain the solutions x = -2 and x = 12.
Conclusion
As demonstrated, we can solve the quadratic equation (x-5)² - 49 = 0 using factoring, the square root property, or the quadratic formula, all leading to the same solutions: x = -2 and x = 12. Choosing the most efficient method depends on the specific form of the equation and your preference.