Solving Quadratics by Graphing
I can solve a quadratic equation from its graph and say how many real solutions it has.
What students learn
Students learn to solve ax² + bx + c = 0 by reading where the graph of y = ax² + bx + c meets the x-axis, find every solution (two, one or none), and verify each one in the equation.
See inside
The key rule
Read the solutions where the parabola crosses or touches the x-axis. It turns, so after one crossing, look for a second.
solutions = x-intercepts
Opens up: vertex below the x-axis, two; on it, one; above it, none.
Watch out
A student solves x² − 6x + 8 = 0 by graphing, finds x = 4 and stops.
The mistake: x = 4One crossing; the parabola turns and crosses again.
The fix: x = 2 or x = 4Check x = 2: 4 − 12 + 8 = 0.
The mistakes it targets
- Finds one crossing and stops
- Drops the negative sign of a crossing left of the y-axis
- Reads the y-intercept as a solution
- Gives the y-values of the crossings instead of the x-values
- Says a vertex above the x-axis means no solution, even opening down
- Gives the vertex, or two solutions, when the parabola only touches the axis
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.