Quadratic Equations: 30 Practice Problems Covering All 6 Core Concepts
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.
Start the 30-Problem Practice Set →
What the Reel Covered — and What You Need to Practice
The six concepts form a complete toolkit for quadratic equations. Here's what each one does and where it trips people 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. Factoring first saves time. These five problems are designed to build that instinct.
What to watch for: x² − 5x + 6 = 0 factors as (x−2)(x−3) = 0. If you went straight for the formula, 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. Understanding it once makes everything else click.
What to watch for: The step where you add (b/2)² to both sides. Getting the sign wrong here is the most common error.
Part 3 — Quadratic Formula
The general-purpose tool. Works on every quadratic.
The formula itself is simple. The mistakes happen inside the discriminant — specifically with negative values of b and c.
What to watch for: 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, you can substitute b = 2b' and simplify the formula to:
x = (−b' ± √(b'² − ac)) / aThis cuts the arithmetic in half. Most students have never used it — which means they're doing unnecessary work on half the problems they see.
What to watch for: 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 > 0 → two distinct real roots
- D = 0 → one repeated root
- D < 0 → two complex (non-real) roots
The five problems here are about conditions — finding what values of a parameter k produce a given root type.
What to watch for: These problems require inequalities, not just equations. Setting D > 0 and solving for k is a different skill from setting D = 0.
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 lets you compute expressions like α² + β² or 1/α + 1/β directly from the coefficients.
What to watch for: 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 test is structured exactly like the six parts above: 5 problems per concept, 30 total.
One question appears at a time. You type your answer — or work it out on paper — and navigate through the set at your own pace. When you submit, every question expands to show the correct answer and a step-by-step solution.
There's no time limit. No account required.
What to Do If You Get 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.
That's the only part of this process that can't be automated.
