%5E2%3D75.jpeg "(3x-1)^2=75")
Solving the Equation: (3x - 1)^2 = 75
This article will guide you through the steps to solve the equation (3x - 1)^2 = 75. We'll use algebraic methods to find the possible values of 'x'.
Understanding the Equation
The equation (3x - 1)^2 = 75 represents a quadratic equation. This means it involves a variable ('x') raised to the power of 2. To solve for 'x', we need to isolate it.
Solving for 'x'
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Take the square root of both sides: √(3x - 1)^2 = ±√75 3x - 1 = ±√75
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Simplify the square root: 3x - 1 = ±5√3
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Isolate 'x': 3x = 1 ± 5√3
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Divide both sides by 3: x = (1 ± 5√3) / 3
Solutions
Therefore, the equation (3x - 1)^2 = 75 has two solutions:
- x = (1 + 5√3) / 3
- x = (1 - 5√3) / 3
Verification
You can verify these solutions by substituting them back into the original equation:
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For x = (1 + 5√3) / 3: (3 * [(1 + 5√3) / 3] - 1)^2 = (1 + 5√3 - 1)^2 = (5√3)^2 = 75
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For x = (1 - 5√3) / 3: (3 * [(1 - 5√3) / 3] - 1)^2 = (1 - 5√3 - 1)^2 = (-5√3)^2 = 75
Both solutions satisfy the original equation, confirming their validity.
Conclusion
By applying algebraic techniques, we have successfully solved the equation (3x - 1)^2 = 75, finding two distinct solutions for 'x'. It's essential to remember that quadratic equations often have multiple solutions, and verifying them is crucial to ensure their accuracy.