x² − 5x + 6 = 0
- 1.Find two numbers with product 6 and sum −5: −2 and −3
- 2.Factor: (x − 2)(x − 3) = 0
x = 2 or x = 3
Verified: 4 − 10 + 6 = 0 ✓ and 9 − 15 + 6 = 0 ✓.
Quadratic equations — factored or formula’d, then proven
Type any quadratic. The solver tries factoring first (like your teacher would), falls back to the quadratic formula when needed, and plugs every root back in to prove it works.
Scope: quadratics in one variable — factoring, the formula, and complex roots. Cubic equations are not covered yet.
- ✓100% free — no paywall, ever
- ✓No sign-up, no app
- ✓Every answer verified before you see it
In short
A quadratic equation solver finds the roots of ax² + bx + c = 0. VerifySolve tries factoring first because it is fastest, falls back to the quadratic formula when factoring fails, and reports complex roots when the discriminant is negative. Every root is then plugged back into the original equation and shown to make it exactly zero.
The quadratic toolkit
Three ideas run every quadratic: standard form, the discriminant’s prediction, and the formula that always works. Factoring is the shortcut you take when the numbers are kind.
Standard form
The quadratic formula
The discriminant
Factored form
Worked examples — every root checked
Notice the pattern: factoring when it is clean, the formula when it is not, and complex roots handled honestly instead of declared “impossible”.
x² + 2x − 8 = 0
- 1.Two numbers with product −8 and sum 2: 4 and −2
- 2.Factor: (x + 4)(x − 2) = 0
x = 2 or x = −4
Verified: 4 + 4 − 8 = 0 ✓ and 16 − 8 − 8 = 0 ✓.
2x² − 7x + 3 = 0
- 1.Discriminant: (−7)² − 4(2)(3) = 49 − 24 = 25
- 2.x = (7 ± √25) / 4 = (7 ± 5) / 4
x = 3 or x = 1/2
Verified: 18 − 21 + 3 = 0 ✓ and 1/2 − 7/2 + 3 = 0 ✓.
x² − 4x + 13 = 0
- 1.Discriminant: 16 − 52 = −36 — negative, so the roots are complex
- 2.x = (4 ± √−36) / 2 = (4 ± 6i) / 2
x = 2 + 3i or x = 2 − 3i
Verified with complex arithmetic: (2 + 3i)² − 4(2 + 3i) + 13 = 0 ✓.
Common mistakes — and how to dodge them
Forgetting that b carries its sign
In x² − 5x + 6, b is −5, so −b is +5. Reading b as “5” is the single most common formula error. Write a, b, c out with signs before touching the formula — the solver does this in its steps.
Dividing only part of the top line by 2a
The fraction bar covers everything: (−b ± √D) is all divided by 2a. Dividing just the root term gives a wrong second answer. Keep the brackets until the final split into two roots.
Giving up when it “doesn’t factor”
Most real quadratics do not factor over the integers — that is normal, not a dead end. The formula always works; switch methods instead of forcing a factor pair that does not exist.
Quadratic Equation Solver — questions
Which method does the solver use — factoring or the formula?
Factoring first, when the quadratic factors neatly over the rationals — because it is faster and teaches more. If it does not factor, the solver switches to the quadratic formula and says so in the steps. That is the same order a good teacher uses.
What does the discriminant tell me?
The discriminant b² − 4ac predicts the roots before you solve: positive means two real roots, zero means one repeated root, negative means two complex roots. The solver computes it as part of the formula method.
Can it handle complex (imaginary) solutions?
Yes. When the discriminant is negative, the solver gives exact complex roots like 2 + 3i and 2 − 3i — and still verifies them by plugging back into the original equation.
What if my quadratic is not in standard form?
Type it as it comes — x² = 3x, or 2x² + 5 = 7x. The solver rearranges it into ax² + bx + c = 0 first and shows that step. Terms can be in any order, and coefficients can be fractions or decimals.
Are the roots exact?
Yes — exact fractions and exact simplified radicals where they occur (for example (7 + √25)/4 is simplified all the way to 3). No decimal rounding anywhere in the answer.
Is the quadratic equation solver free?
Completely. No sign-up, no app, and the steps are never paywalled — including the verification.
Keep solving
Guide: How to Solve Quadratic Equations →
All three methods — factoring, formula, completing the square — with fully worked examples.
Factoring Calculator →
Factor trinomials and expressions step by step, including the AC method.
Algebra Calculator →
Linear equations, systems, simplifying and factoring in one box.
Equation Solver →
Any equation type — routed to the right method automatically.
A note on honest use: VerifySolve is built for learning. Use the steps to understand the method, then close the tab and try one on your own — that is where the grade actually comes from. If your teacher or school has a policy on calculators, follow it.