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Solving the Equation: (x+6)^(1/2) = x
This article will explore the process of solving the equation (x+6)^(1/2) = x. This equation involves a square root, which requires specific steps to isolate the variable.
Understanding the Equation
The equation (x+6)^(1/2) = x represents the square root of the expression (x+6) being equal to x. To solve for x, we need to eliminate the square root.
Solving for x
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Square both sides: To eliminate the square root, we square both sides of the equation.
(x+6)^(1/2) * (x+6)^(1/2) = x * x
This simplifies to: x + 6 = x^2
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Rearrange the equation: Move all terms to one side to obtain a quadratic equation.
x^2 - x - 6 = 0
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Factor the quadratic equation: Factor the quadratic equation to find the roots.
(x-3)(x+2) = 0
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Solve for x: Set each factor equal to zero and solve for x.
x - 3 = 0 => x = 3 x + 2 = 0 => x = -2
Checking for Extraneous Solutions
It's important to check if the solutions obtained satisfy the original equation. Substituting x = 3 into the original equation:
(3 + 6)^(1/2) = 3 (9)^(1/2) = 3 3 = 3 (This solution is valid)
Substituting x = -2 into the original equation:
(-2 + 6)^(1/2) = -2 (4)^(1/2) = -2 2 = -2 (This solution is not valid)
Conclusion
Therefore, the only valid solution for the equation (x+6)^(1/2) = x is x = 3. The solution x = -2 is extraneous as it does not satisfy the original equation. It's crucial to check for extraneous solutions when dealing with equations involving square roots.