Quadratic Formula Calculator
x = (−b ± √(b²−4ac)) / 2a with the discriminant doing the talking: rational, irrational, repeated or complex — each case solved, classified and checked by Vieta.
Quadratic Formula Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- Degree exactly 2 — a = 0 is refused as linear, with the linear root printed
- Roots printed to 6 working places; no radical-form simplification beyond the perfect-square case
In short: x² − 5x + 6 = 0: the discriminant is (−5)² − 4·1·6 = 1, a positive perfect square, so both roots are rational: x = (5 ± 1)/2 gives 3 and 2, and the formula even shows the factorization (x − 3)(x − 2). Vieta agrees: the roots sum to 5 = −b/a and multiply to 6 = c/a. The vertex sits at x = 2.5, dead between the roots.
Formula
x = (−b ± √D) / 2a · D = b² − 4ac
D > 0: two real · D = 0: one repeated · D < 0: two complex
the discriminant classifies before the formula computes — count the roots first, then find them.
Worked Example
- Discriminant. D = b² − 4ac — the count and the type of the roots live here, before any division.
- Formula. x = (−b ± √D)/2a — with D negative the square root goes imaginary and the roots become conjugates: −b/2a ± √|D|/2a · i.
- Verify. Vieta’s checks are printed: the roots must sum to −b/a and multiply to c/a — recomputed live from the roots just found.
The four cases on real coefficients: D a positive perfect square → two rational roots (and a factorization over the integers); D positive otherwise → two irrational roots; D = 0 → one repeated root at −b/2a; D negative → two complex conjugates, no real x-intercepts.
Strengths & Limits Of This Model
Where this engine is strong
- The case classification is printed before the roots, so the answer type is never a surprise
- Vieta checks recomputed live — the page audits itself on every drive
Where it stops
- No completing-the-square walkthrough
- No graph
Practical Use Cases
Projectile peaks
the vertex card is the peak height question in disguise
Factoring check
when D is a perfect square the integer factorization falls out
Curve sketching
roots, axis of symmetry and vertex in one pass
Methodology & Editorial Standards
D = b² − 4ac in exact arithmetic. D > 0: roots (−b ± √D)/2a; a perfect-square D additionally prints the integer factorization. D = 0: the repeated root −b/2a. D < 0: the conjugate pair −b/2a ± √|D|/2a·i. Vieta’s sum and product are recomputed from the printed roots as a live check. a = 0 refuses the formula and prints the linear root −c/b (or the b = 0 verdict).
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Quadratic Formula Calculator — 9 Expert FAQs
9 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
What does the discriminant actually discriminate?
The number and type of solutions. D = b² − 4ac sits under the square root in the formula, so its sign decides everything: positive means two real roots, zero means one repeated root, negative means none on the real line — just a conjugate pair in the complex numbers. It classifies before it computes, which is why the page prints it first.
Why does a perfect-square discriminant matter?
Because √D is then a whole number, so both roots are rational — and the quadratic factors over the integers: x² − 5x + 6 has D = 1, roots 3 and 2, and indeed (x − 3)(x − 2) = x² − 5x + 6. Factoring by sight is just the discriminant test done in your head.
What are the roots when D is negative?
A conjugate pair: −b/2a ± (√|D|/2a)·i. For x² + x + 1 (D = −3) that is −0.5 ± 0.866…i. There are no real x-intercepts — the parabola never touches the axis — but the complex roots are as exact as any other, and they always arrive in mirrored pairs because the formula only differs by the ±.
Why is D = 0 called a REPEATED root?
Both branches of the ± collapse onto the same value: x = −b/2a twice. The parabola touches the x-axis at exactly one point — the vertex sits ON the axis — and the polynomial factors as a(x − r)². 2x² − 4x + 2 = 2(x − 1)² is the shape.
What if a is zero?
Then there is no quadratic and no formula: ax² + bx + c becomes bx + c, a straight line, and dividing by 2a would divide by zero. The page refuses the pretense but still helps: it prints the linear root x = −c/b clearly labeled as linear. If b is zero too, there is no x at all — either nothing or everything, and the page says which.
What do Vieta’s formulas add?
A checksum that costs nothing. The roots of any quadratic sum to −b/a and multiply to c/a, so after computing x₁ and x₂ the page recomputes both identities from the roots themselves. On x² − 5x + 6: 3 + 2 = 5 and 3 × 2 = 6 — the arithmetic witnesses itself. If your hand-computed roots fail Vieta, no graph is needed to convict them.
How does the vertex relate to the roots?
The parabola is symmetric about x = −b/2a, so the vertex sits exactly halfway between the roots — 2.5 between 2 and 3 on the default. That symmetry is WHY Vieta’s sum is −b/a: the two roots hang at equal distances either side of the axis. The vertex card prints the axis and the roots’ midpoint together so the geometry is visible.
Can I enter decimals or negatives?
All coefficients accept any real number — decimals, negatives, zero (for b and c; a = 0 gets the linear refusal). The discriminant arithmetic is exact subtraction and multiplication; the square root and divisions are double-precision, and the printed roots carry the same six-place honesty as the rest of the site.
Why print the factored form only sometimes?
Because it only exists over the integers when D is a positive perfect square (and a divides nicely). Printing (x − 3)(x − 2) for x² − 5x + 6 is honest; printing a fake factorization for x² − 4x − 3 (D = 28, irrational roots) would be fabrication. The page shows the form the case actually supports — irrational and complex cases get their own honest portraits.