Completing the Square to Find the Vertex
I can complete the square to rewrite a quadratic in vertex form and read off the vertex.
The problem this sheet solves
Every earlier move in the course preserved an expression by doing the same thing to both sides of an EQUATION. Here there is no equation — an expression is being rewritten — so there is no other side to balance against, and the only way to stay honest is to add and subtract in the same place. That is a genuinely new kind of step, and a warning to be careful will not install it. The sheet names the step as new rather than warning about it, and hands over a one-line check: expand back, or substitute a single value. At x = 0 the original is 5 and the un-subtracted version is 14. It also handles the case students meet next and are never shown — when the leading coefficient is not 1, the amount taken back is a times the square, not the square.
The key rule
Add (b/2)² to complete the square and subtract the same amount in the same line. With a in front, what comes back out is a times that square.
In and out, same line.
Watch out
The mistake: A student writes x2 + 6x + 5 = (x + 3)2 + 5.
The fix: (x + 3)2 − 4. Adding 9 to complete the square CHANGED the expression, so the 9 has to come back out. Expand theirs: it gives x2 + 6x + 14.
What is included
- Reference page — student-friendly definitions, a worked example, a Watch Out error pair, and the key rule
- Practice page — six problems, building from foundational to multi-step
- Apply page — a word problem, a reasoning prompt, and an error-analysis task
- Assess — a half-page exit ticket, two to a sheet, with a score box
- Full answer key — every item worked, plus a grading guide that says what each wrong answer tells you
- Ink-saver edition of all 7 pages
- 14-slide teacher deck (PowerPoint) with presenter notes on every slide
Standard codes refer to the Common Core State Standards for Mathematics. Math Class 678 is an independent publisher, not affiliated with, sponsored by, or endorsed by NGA Center or CCSSO. Every product is sold on Teachers Pay Teachers.