Complete the square for \(x^{2}-7x+1\) without decimal approximations. Show the fractional half-coefficient, its square, the constant adjustment, and the resulting vertex.

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Half of −7 is −7/2, so add and subtract −7/2, giving (x − 7/2)² − 5/2. The square of −7/2 is 49/4, not −7/2.

Half of −7 is −7/2, so add and subtract 49/4: x² − 7x + 49/4 − 49/4 + 1 = (x − 7/2)² − 45/4. The vertex is (7/2, −45/4).

(x − 7/2)² − 45/4 expands correctly, but the vertex is (−7/2, −45/4). The vertex x-coordinate is 7/2.

Add 49/4 to form (x − 7/2)² and keep 1, giving (x − 7/2)² + 1. Adding 49/4 requires subtracting 49/4.

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Curriculum context

Course: Math II
Standard: A-SSE.3.b
Category: Algebra
Domain: Seeing Structure in Expressions
Objective: Complete the square to reveal maximum or minimum values of quadratic functions.
Problem type: Complete the square for a monic quadratic