(x+6)^1/2=x

3 min read Jun 17, 2024
(x+6)^1/2=x

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

  1. 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

  2. Rearrange the equation: Move all terms to one side to obtain a quadratic equation.

    x^2 - x - 6 = 0

  3. Factor the quadratic equation: Factor the quadratic equation to find the roots.

    (x-3)(x+2) = 0

  4. 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.