Final answer:
To solve the equation √2x + 29 - 7 = x, we isolate the square root term, eliminate the square root, and factor the resulting quadratic equation. The solutions are x = 22 and x = -22. However, after checking for extraneous solutions, we find that x = -22 is an extraneous solution.
Step-by-step explanation:
To solve the equation √2x + 29 - 7 = x, we first isolate the square root term by subtracting x from both sides: √2x + 29 - 7 - x = 0. Simplifying this equation gives us √2x - x = -22. Combining like terms, we have √2x - x = -22.
To eliminate the square root, we square both sides of the equation: (√2x - x)^2 = (-22)^2. Expanding and simplifying the equation yields 2x - 2√2x + x^2 = 484.
Rearranging the equation in standard quadratic form, we get x^2 - 2√2x + 2x - 484 = 0. Simplifying further, we have x^2 - 2√2x + 2x - 484 = 0.
Factoring the quadratic equation gives us (x - 22)(x + 22) = 0. Therefore, the solutions to the equation are x = 22 and x = -22. However, we need to check for extraneous solutions to ensure the validity of the solutions.
Substituting x = 22 back into the original equation, we have √2(22) + 29 - 7 = 22. Simplifying this equation gives us 33 + 29 - 7 = 22. Since both sides of the equation are equal, x = 22 is a valid solution.
Substituting x = -22 back into the original equation, we have √2(-22) + 29 - 7 = -22. Simplifying this equation gives us -33 + 29 - 7 = -22. Since the right side of the equation is not equal to the left side, x = -22 is an extraneous solution.
Learn more about extraneous solution