Question

Which comparison of the two equations is accurate?

Equation A: squareroot x^2 +3x-6 =squareroot x+2
Equation B: ^3squareroot x^2 +3x -6 = ^3 squareroot x+2
Both equations have different solutions because the square root of a number is not the same as the cube root of a number.
Both equations result in different equations after eliminating the radicals, so they have the different solutions.
Both equations have the same potential solutions, but equation A might have extraneous solutions.
Both equations have the same potential solutions, but equation B might have extraneous solutions.

Answer 1

Correct choice: "Both equations have the same potential solutions, but equation A might have extraneous solutions."

Reason:
- If you eliminate the radicals by taking the appropriate power, both give the same algebraic equation
x^2 + 3x - 6 = x + 2 → x^2 + 2x - 8 = 0 → (x+4)(x-2) = 0, so the potential solutions are x = -4 and x = 2.
- For equation B (cube roots) the real cube root is one-to-one and defined for all real numbers, so both x = -4 and x = 2 are valid solutions.
- For equation A (square roots) the principal square root requires the radicands to be ≥ 0. x = -4 makes x+2 = -2 and x^2+3x-6 = -2, so the square roots are not defined there; x = 2 is valid. Thus x = -4 is extraneous for A.