If x is a real number, is x equal to 3?
(1) x³ = 9x
(2) x² = 3x
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Answer & Explanation
Correct answer
E
Never divide an equation by a variable that could be zero; factor instead.
(1) alone: x³ = 9x becomes x³ − 9x = 0, so x(x² − 9) = 0, giving x = 0, x = 3, or x = −3. That includes a yes (x = 3) and a no (x = 0), so it is insufficient. (2) alone: x² = 3x becomes x² − 3x = 0, so x(x − 3) = 0, giving x = 0 or x = 3, again a yes and a no: insufficient. Together: the common solutions of both equations are the values in {0, 3, −3} that also lie in {0, 3}, namely x = 0 and x = 3. Since x could still be 0 (answer no) or 3 (answer yes), even both statements together do not settle the question.
The trap is the strong pull to divide each equation by x. Canceling x in statement (2) leaves x = 3, which makes (2) look sufficient on its own and pulls you toward (B); canceling in statement (1) leaves x² = 9, so x = 3 or x = −3. In every one of these shortcuts the root x = 0 has been thrown away, yet it survives both original equations, and that overlooked root is exactly why neither statement alone nor both together can settle whether x = 3. Answer: (E).