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Solving the Equation (x - 4)^2 = 9
This article will guide you through the process of solving the equation (x - 4)^2 = 9. We'll explore the steps involved and demonstrate how to arrive at the solution.
Understanding the Equation
The equation represents a quadratic expression where the left-hand side is a perfect square trinomial. This makes solving the equation relatively straightforward.
Solving for x
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Take the square root of both sides: √((x - 4)^2) = ±√9
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Simplify: x - 4 = ±3
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Isolate x: x = 4 ± 3
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Calculate the two possible solutions:
- x = 4 + 3 = 7
- x = 4 - 3 = 1
The Solutions
Therefore, the solutions to the equation (x - 4)^2 = 9 are x = 7 and x = 1.
Verification
To verify our solutions, we can substitute them back into the original equation:
- For x = 7: (7 - 4)^2 = 3^2 = 9 (verified)
- For x = 1: (1 - 4)^2 = (-3)^2 = 9 (verified)
Both solutions satisfy the original equation, confirming their validity.
Conclusion
Solving the equation (x - 4)^2 = 9 involves taking the square root of both sides and simplifying the resulting expression. This leads to two possible solutions: x = 7 and x = 1. By substituting these solutions back into the original equation, we can verify their accuracy.