Advanced Math practice question

Answer: A. x = −1 and x = 7

Add 32: 2(x − 3)² = 32. Divide by 2: (x − 3)² = 16. Take square roots: x − 3 = ±4, so x = 3 + 4 = 7 or x = 3 − 4 = −1. Check: 2(7 − 3)² − 32 = 2(16) − 32 = 0 and 2(−1 − 3)² − 32 = 2(16) − 32 = 0. (B) comes from mishandling the sign of the binomial, writing x = −3 ± 4 instead of x = 3 ± 4. (C) comes from keeping only the principal square root (x − 3 = 4) and dropping the negative root, so one solution is lost. (D) comes from taking the square root before dividing by 2, i.e. solving (x − 3)² = 32 and getting x = 3 ± 4√2.

Why the other answers are wrong

B. x = 1 and x = −7

You mishandled the sign of the binomial, writing x = −3 ± 4 instead of x = 3 ± 4; since x − 3 = ±4, you add 3 to both sides.

C. x = 7

You kept only the principal square root (x − 3 = 4) and dropped x − 3 = −4, losing the solution x = −1.

D. x = 3 − 4√2 and x = 3 + 4√2

You took the square root before dividing by 2, solving (x − 3)² = 32; divide by 2 first so that (x − 3)² = 16 and the roots are whole numbers.

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