%5E2%3D64.jpeg "(x-7)^2=64")
Solving the Equation: (x - 7)² = 64
This equation presents a simple yet effective demonstration of solving quadratic equations. Here's how we can break it down:
Understanding the Equation
The equation (x - 7)² = 64 represents a quadratic equation. It involves a variable (x) raised to the power of two. To solve for x, we need to isolate it.
Solving the Equation
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Taking the Square Root: Start by taking the square root of both sides of the equation. Remember that taking the square root results in both positive and negative values: √(x - 7)² = ±√64
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Simplifying: Simplify the square roots: x - 7 = ±8
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Isolating x: Add 7 to both sides of the equation to isolate x: x = 7 ± 8
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Finding the Solutions: Calculate the two possible solutions:
- x = 7 + 8 = 15
- x = 7 - 8 = -1
Verifying the Solutions
To verify our solutions, substitute each value of x back into the original equation:
- For x = 15: (15 - 7)² = 8² = 64
- For x = -1: (-1 - 7)² = (-8)² = 64
Both solutions satisfy the original equation, confirming their validity.
Conclusion
Therefore, the solutions to the equation (x - 7)² = 64 are x = 15 and x = -1. This problem highlights the importance of considering both positive and negative roots when dealing with square roots in quadratic equations.