Quadratic Equations Practice Problems — 30 Problems Across 6 Concepts
30 quadratic practice problems across 6 concepts: factoring, completing the square, quadratic formula, even-coefficient formula, discriminant, and Vieta's formulas. Step-by-step solutions on submit.
TL;DR
30 quadratic practice problems across 6 concepts: factoring, completing the square, quadratic formula, even-coefficient formula, discriminant, and Vieta's formulas. Each problem shows a step-by-step solution on submit — no account required. Start the practice set →
You watched the reel.
You nodded along — the discriminant, the quadratic formula, Vieta's formulas — it all made sense while someone else was explaining it.
Then you sat down with a blank page and hit a wall.
That gap between watching and solving is exactly what these 30 problems are for.
The 6 Quadratic Concepts — and Where Each One Trips Students Up
Part 1 — Factoring
The fastest method when it works. The problem is knowing when it works.
Most students reach for the quadratic formula even when the equation factors cleanly. These five problems are designed to build that instinct.
x² − 5x + 6 = 0 factors as (x−2)(x−3) = 0. If you went straight for the formula here, that's the habit to break.
Part 2 — Completing the Square
This is the method students skip — and then can't derive the quadratic formula when asked to.
Completing the square is not just a backup method. It is the reason the quadratic formula exists.
The most common error: the step where you add (b/2)² to both sides. Getting the sign wrong here cascades into every line that follows.
Part 3 — Quadratic Formula
The general-purpose tool. Works on every quadratic equation.
The formula itself is simple. The mistakes happen inside the discriminant — specifically with negative values of b and c.
2x² + 3x − 1 = 0 gives D = 9 + 8 = 17, not 9 − 8. The negative c adds to the discriminant.
Part 4 — Even-Coefficient Formula
When b is even, substitute b = 2b' and simplify:
x = (−b' ± √(b'² − ac)) / a
This cuts the arithmetic in half. Most students have never used it — which means they're doing unnecessary work on half the quadratic formula problems they encounter.
x² − 4x + 2 = 0 → b' = −2, D' = 4 − 2 = 2, x = 2 ± √2. Compare that to running the full formula.
Part 5 — Discriminant
D = b² − 4ac tells you the nature of the roots before you solve anything.
| D value | Root type |
|---|---|
| D > 0 | Two distinct real roots |
| D = 0 | One repeated real root |
| D < 0 | Two complex (non-real) roots |
These five problems are about conditions — finding what values of a parameter k produce a given root type. They require setting up inequalities, not just equations.
Part 6 — Vieta's Formulas
For ax² + bx + c = 0 with roots α and β:
| Formula | |
|---|---|
| Sum of roots | α + β = −b/a |
| Product of roots | αβ = c/a |
You never need to find the roots individually. Vieta's formulas let you compute expressions like α² + β² or 1/α + 1/β directly from the coefficients.
α² + β² = (α+β)² − 2αβ — this identity, and variations of it, appears in every problem in this section.
How the Practice Set Works
The set is structured exactly like the six parts above: 5 problems per concept, 30 total.
One question appears at a time. You work it out on paper, enter your answer, and submit. Every question then expands to show the correct answer and a full step-by-step solution.
No time limit. No account required. Works on any device.
Open the 30-Problem Practice Set →
What to Do When You're Stuck
Don't scroll to the solution immediately.
The moment right before you give up — when you're stuck but still thinking — is where the learning happens. Sit with it for two minutes. Try a different approach. Then check the solution and trace back to where your path diverged from the correct one.
That's the only part of this process that can't be automated.
Frequently Asked Questions
Which quadratic method should I use on the SAT?
Factoring first if the equation factors cleanly — it's faster. Use the quadratic formula when factoring isn't obvious. The even-coefficient shortcut applies whenever b is even and saves one step in the discriminant calculation.
How is the discriminant used in SAT problems?
SAT discriminant problems typically ask for the number of real solutions or the value of a parameter that produces a specific root type. Set D > 0 for two distinct roots, D = 0 for exactly one root, D < 0 for no real roots, then solve for the parameter.
What are Vieta's formulas used for on the SAT?
Vieta's formulas appear when a problem gives you a quadratic and asks for expressions involving both roots — like their sum, product, or α² + β² — without requiring you to find each root individually.
Is this practice set based on actual SAT questions?
The 30 problems cover all six concepts that appear in SAT Math quadratic questions, structured to match the difficulty range from Easy to Hard. The interface replicates the Bluebook format used on the Digital SAT.
Last updated: 2026-05-29
