Quadratic Equations and the Parabola
▶︎ Watch it animatedinteractive step-through · ~3 min · optionalQuadratics show up two ways on the SAT, as an equation to solve and as a parabola to read, and the same three coefficients carry both. The algebra is short. The points leak in four moves: a vertex form read with the wrong sign, the discriminant turned into the wrong count of solutions, a piece of the quadratic formula or completing the square dropped along the way, and the right number handed in as the wrong feature of the parabola.
These patterns aren't really about whether you can solve a quadratic. They're about reading vertex form in the right direction, letting the sign of the discriminant set the count, carrying every piece of the formula, and reporting the feature the item actually asked for, the root, the vertex, an intercept, or a maximum, not the one sitting next to it.
The work
4 ways in · any order
Lesson
Quadratic Equations and the Parabola
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Read vertex form in the right direction, let the discriminant tell you how many solutions there are, solve cleanly by formula or completing the square, then report the feature the item names. The lesson works the method and the four patterns that derail it, and it saves the trap for last: the right number for a feature the item never asked about.
Diagnostic
10-item topic check
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Ten items across the four patterns: misreading vertex form, misreading the discriminant, a slip in the quadratic formula or completing the square, and reporting the wrong feature of the parabola. Take it cold to surface the ones still catching you, or after the lesson to confirm they are gone.
Grid-in Check
Student-produced response
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About a quarter of SAT math answers are typed, not chosen, with no options to react to. These grid-in items diagnose by the value you enter, then route into the same drills the multiple-choice check feeds.
Targeted Practice
Drill a single pattern
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Pick one of the failure modes you've missed and grind it on its own. The round is adaptive: two correct in a row clears the pattern and you move on.