If a = 0.1039, evaluate √(4a² − 4a + 1) + 3a.

Problem restatementCompute √(4a² − 4a + 1) + 3a for a = 0.1039.

Concept/ApproachRecognize a perfect-square trinomial: 4a² − 4a + 1 = (2a − 1)². Then √( (2a − 1)² ) = |2a − 1|.

Step-by-step calculation2a − 1 = 2(0.1039) − 1 = 0.2078 − 1 = −0.7922|2a − 1| = 0.79223a = 3 × 0.1039 = 0.3117Sum = 0.7922 + 0.3117 = 1.1039

Verification/AlternativeAlgebraic shortcut: |2a − 1| + 3a = (1 − 2a) + 3a = 1 + a, since 2a − 1 < 0. With a = 0.1039 ⇒ 1 + a = 1.1039.

Common pitfallsDropping the absolute value when taking the square root of a square.

Final Answer1.1039