Question

Which comparison of the two equations is accurate?

Equation A: \sqrt{x^{2}+3x-6}=\sqrt{x+2}
Equation B:3\sqrt{x^{2}+3x-6}=3\sqrt{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

None of the four given choices is correct.

Reason: Equation B is just 3 times equation A, and multiplying both sides of an equation by a nonzero constant (3) does not change its solution set. So A and B are equivalent and have the same potential and extraneous solutions.

Solve (by squaring):
x^2 + 3x - 6 = x + 2
x^2 + 2x - 8 = 0
(x - 2)(x + 4) = 0 → x = 2 or x = −4.

Check in the original (radical) equation: x must make both radicands nonnegative. x = 2 works; x = −4 gives a negative radicand and is extraneous. Thus both A and B have the same legitimate solution x = 2.