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Math · Algebra

Quadratic equations

Solve by factoring, completing the square, or the quadratic formula — and know which method is fastest for the equation in front of you. Every question comes with a written walkthrough of the method used and why it fit.

Practise quadratic equations nowSee sample questions
3
Difficulty tiers
45s
Target pace
50%
Average first-attempt score

What's covered

  • Solving by factoring — set the equation to zero, factor, then apply the zero-product property to find both roots.
  • The quadratic formula — apply x = (−b ± √(b² − 4ac)) / 2a when factoring isn't fast or possible, tracking each sign carefully.
  • Completing the square — convert to vertex form to solve or to identify the vertex directly, without needing to factor.
  • The discriminant — use b² − 4ac to predict the number and type of solutions before solving the whole equation.
  • Word problems that reduce to a quadratic — recognize when a scenario — area, projectile motion — produces a quadratic, and discard any root that doesn't make sense in context.

Where students lose marks

Forgetting the ± in the quadratic formula

A quadratic almost always has two roots; dropping the minus branch loses half the answer.

Not setting the equation to zero before factoring

Factoring only works when one side of the equation is zero — factor the wrong side and the zero-product property doesn't apply.

Keeping an extraneous solution in a context problem

A negative length or negative time is mathematically valid but physically meaningless — the context, not the algebra, tells you to discard it.

Three sample questions

Straight from the bank — one per difficulty tier. Reveal the answer to see the explanation you'd get in a real session.

Sample 01Foundation

Solve by factoring: x² − x − 6 = 0

Sample 02Core

Solve using the quadratic formula: 2x² + 3x − 2 = 0

Sample 03Advanced

A ball is launched upward with height h(t) = −16t² + 64t + 5 (feet, seconds). At what time, to two decimals, does it hit the ground?

How to practise this

Quadratics reward knowing which method to reach for — factoring when it's fast, the formula when it isn't — practice builds that judgment as much as the mechanics.

Try factoring first, formula second

For every equation, spend five seconds checking whether it factors cleanly before defaulting to the quadratic formula. Untimed practice is where that judgment call gets fast.

Then 45 seconds a question

Move to timed sessions once method selection is quick. Quadratics run longer than linear equations — budget the extra time.

Fold into a mock

Combine quadratic equations with quadratic functions in a mock — the algebra here is the same skill tested from a graphing angle there.

Stop defaulting to the formula.
Start choosing the fastest method.

  • A method-selection walkthrough on every item
  • Timed or untimed sessions, any length you like
  • Tracked separately — see your quadratic equations accuracy and pace over time
Start practising free

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