Completing the Square
Rewrites a quadratic in vertex form.
Formula
Variables
Example
x² + 6x + 5 = (x+3)² − 4, so the vertex is at x = −3.
How it works
Completing the square turns a quadratic expression into a squared binomial plus a constant, which makes the parabola’s turning point easy to see. It works because the identity (x + p)^2 = x^2 + 2px + p^2 lets us rewrite the x-term and then compensate by adding or subtracting the same amount so the value stays unchanged. In the general form ax^2 + bx + c, factoring out a first creates a perfect-square pattern inside the brackets.
When to use it
Use this when you need the vertex of a parabola, want to rewrite a quadratic in vertex form, or need to solve a quadratic equation by taking square roots. It is also useful in graphing and in calculus when analyzing the shape and minimum/maximum of a quadratic.
Worked example, step by step
Rewrite 2x^2 + 8x + 3 by completing the square. Step 1: Factor out 2 from the x-terms: 2(x^2 + 4x) + 3 Step 2: Take half of 4, which is 2, and square it: 2^2 = 4 Step 3: Add and subtract 4 inside the brackets: 2(x^2 + 4x + 4 - 4) + 3 Step 4: Group the perfect square: 2((x + 2)^2 - 4) + 3 Step 5: Distribute the 2: 2(x + 2)^2 - 8 + 3 Final answer: 2x^2 + 8x + 3 = 2(x + 2)^2 - 5
Common mistakes to avoid
- Forgetting to factor out a when the x^2 coefficient is not 1
- Adding the square term but not subtracting it, which changes the expression
- Using b/2 instead of b/(2a) when completing the square in ax^2 + bx + c
Did You Know?
Completing the square is how the quadratic formula was originally derived — geometrically, you literally complete a square shape.
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