%5E2%3D25.jpeg "(x-1)^2=25")
Solving the Equation (x-1)^2 = 25
This equation presents a simple yet important example of solving quadratic equations. Let's break down the process step by step:
Understanding the Equation
The equation (x-1)^2 = 25 represents a quadratic equation, meaning it involves a variable raised to the power of 2. We need to find the values of 'x' that satisfy this equation.
Solving for 'x'
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Take the square root of both sides:
√((x-1)^2) = ±√25
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Simplify:
x - 1 = ±5
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Isolate 'x':
x = 1 ± 5
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Solve for both possible solutions:
- x = 1 + 5 = 6
- x = 1 - 5 = -4
Therefore, the solutions to the equation (x-1)^2 = 25 are x = 6 and x = -4.
Verifying the Solutions
To verify our solutions, we can substitute each value of 'x' back into the original equation:
- For x = 6: (6 - 1)^2 = 5^2 = 25 (True)
- For x = -4: (-4 - 1)^2 = (-5)^2 = 25 (True)
Both solutions satisfy the original equation, confirming our calculations.
Conclusion
Solving the equation (x-1)^2 = 25 demonstrates the process of solving quadratic equations through taking square roots and isolating the variable. We found two solutions, x = 6 and x = -4, which both satisfy the original equation. This type of problem emphasizes the importance of understanding the properties of squares and square roots in solving mathematical equations.