Completing the square
SAT glossary · Math
Definition
Rewriting a quadratic as a(x-h)2 +k to reveal the vertex, or a circle's equation to reveal center and radius. The trap in circle problems is reading the loose constant as r2 without completing the square.
Completing the square in plain English
Completing the square is a technique for rewriting a quadratic expression like ax^2+bx+c into the form a(x-h)^2+k, which immediately reveals the vertex (the peak or valley point) of the parabola without needing to graph anything. The same trick works on circle equations too — it reorganizes a messy equation into the clean form (x-h)^2+(y-k)^2=r^2, revealing the circle's center (h,k) and its radius r. The classic trap here is misreading the loose number sitting by itself on one side of a circle equation as if it WERE the radius — it's actually r^2, so you must take its square root to get the true radius, and skipping that step (or skipping completing the square entirely) leads straight to a wrong answer.
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Practice Completing the square
The function f is defined by f(x)=3x^2-12x+7. What is the minimum value of f(x)?
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