Skill 4 of 12

Complete the square to write vertex form

Rewrite $ax^2+bx+c$ in the form $a(x-h)^2+k$ by completing the square.

Common Core standards: A-SSE.3bF-IF.8a
Builds on: Factor quadratic expressions

Common mistakes

What students most often get wrong with this skill:

Adding $(b/2)^2$ without subtracting it to keep the expression equal
Using $b/2$ instead of $(b/2)^2$
Not factoring out $a$ before completing the square when $a \neq 1$
Sign error on $h$ , e.g. writing $(x+3)^2$ as having $h=3$

A worked example

From the lesson, with every step explained.

Example 1: monic, even b

Problem: Write $x^2+6x+2$ in vertex form.

Plan: Here $a=1$ and $b=6$ is even. Find half of $b$ , square it, add and subtract it.

1) Half of $b$ : $6/2=3$ .
2) Square it: $3^2=9$ .
3) Add and subtract $9$ so the value stays the same:

x^2+6x+9-9+2

4) The first three terms are a perfect square (the rule $(x+p)^2=x^2+2px+p^2$ with $p=3$ ):

(x+3)^2-9+2

5) Combine the numbers: $-9+2=-7$ .

Answer:

x^2+6x+2=(x+3)^2-7

Check: expand the answer.

(x+3)^2-7=x^2+6x+9-7=x^2+6x+2

This matches the original. The vertex is $(-3,-7)$ .

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