If x is a positive real number and x − √x = 1, what is the value of x?
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Answer & Explanation
Correct answer
D
Let u = √x, so u ≥ 0 and x = u². The key move is the substitution: x − √x = 1 becomes u² − u − 1 = 0. By the quadratic formula, u = (1 ± √5)/2. The domain restriction u ≥ 0 forces the plus sign, since the negative root (1 − √5)/2 is about −0.62. So u = (1 + √5)/2.
Now track what the question actually asks: it wants x, not √x, so square the result. x = u² = ((1 + √5)/2)² = (1 + 2√5 + 5)/4 = (6 + 2√5)/4 = (3 + √5)/2 ≈ 2.62. Quick check: √x ≈ 1.62, and 2.62 − 1.62 = 1, as required. The choices are irrational surds, so you cannot back-test them by inspection; verifying one means taking the square root of a surd, which is just the forward work again. So the answer is D.
Among the traps: A squares the discarded negative root (1 − √5)/2; B sign-flips the linear coefficient and solves u² + u − 1 = 0; C correctly finds u but reports it without squaring to get x; E uses u² = u + 1 to get x = u + 1 but then adds the original 1 a second time, reaching u + 2 = (5 + √5)/2.