Solving Quadratics by Completing the Square
I can solve a quadratic equation by completing the square on both sides.
What students learn
Students learn to solve a quadratic equation by completing the square: they get x² + bx = c, add (b/2)² to both sides, take square roots (keeping ±), and give exact answers in simplest radical form.
See inside
The key rule
With x² + bx = c, add (b/2)² to both sides. The left side becomes (x + b/2)², a perfect square.
x² + bx + (b/2)² = c + (b/2)²
Then take ± roots (if the right side is ≥ 0) and find x; a negative right side means no real solution. If a ≠ 1, divide every term by a first.
Watch out
A student adds 4 to only the left side of x² − 4x = 5.
The mistake: (x − 2)² = 5The 4 went on the left only.
The fix: x = 5 or x = −1x² − 4x + 4 = 5 + 4, so (x − 2)² = 9.
The mistakes it targets
- Adds (b/2)² to the left side only
- Adds half of b without squaring it
- Loses the sign of half of a negative b
- Subtracts the wrong way when finishing the solution
- Divides only the x terms by the leading coefficient
- Leaves a radical unsimplified, or keeps a negative length in a story
The interactive Teacher Deck (32 slides, PowerPoint)
- The Learn page taught on screen: 8 fill-in blanks, each filled on its own click as students write it
- The worked example one slide per step (four steps), each with the reason for the step
- Six practice and three apply problems: a your-turn slide with a think-time bar, then the solution one line per click
- Jump buttons on every slide: go straight to any problem, the answer bank or the exit ticket
- Speaker notes on each problem: the question to ask, the worked answer and the common error to listen for
- A completed deck PDF (19 pages) with every slide fully built, to print or post
The student packet (9 pages)
- Learn: guided notes with four definitions, the key rule, a worked example and a Watch Out error pair
- Practice: two warm-up, two standard and two stretch problems, with an answer bank for self-checking that hides three common-mistake decoys
- Apply: one word problem, a reasoning prompt and one error analysis
- Assess: an exit ticket of four items, two per page, with an I-can self-rating
- Homework (page 5): eight new problems, two warm-up, four standard and two stretch
- Answer key (pages 6–7): every problem worked step by step, six common mistakes with what to do about each, and a grading guide, plus the homework key (page 8)
- Color and ink-saver editions of the whole packet
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.