How to Complete the Square
Completing the square is the algebraic trick behind the quadratic formula itself. The idea: force part of the expression into a perfect square (x + d)², then tidy up whatever is left over.
The Three Steps
- Normalize: if a ≠ 1, factor a out of the x-terms first.
- Halve and square: take half of the x-coefficient, square it — that is the number that completes the square.
- Balance: add it inside the brackets and subtract the equivalent outside, then collapse to (x + d)² + k.
Reading the Vertex
In y = a(x − h)² + k, the vertex is (h, k). Watch the sign: (x + 3)² means h = −3. If a > 0 the parabola opens upward and the vertex is a minimum; if a < 0 it opens downward.
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