EVERYDAY MATH
Quadratic equation calculator
Solve ax² + bx + c = 0, classify the discriminant, calculate real or complex roots, and show the vertex, axis of symmetry and numerical substitution residuals.
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Enter the coefficients in ax² + bx + c = 0.
Two distinct real roots
3Second root: 2- Discriminant
- 1
- Vertex
- (2.5, -0.25)
- Axis of symmetry
- x = 2.5
- First substitution residual
- 0
- Second substitution residual
- 0
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Verification and calculation details
- Purpose
- Quadratic equation calculator
- Variables and units
- Use the labelled fields and units shown in the calculator. Results use those units unless a conversion is stated.
- Formula and method
- The exact formula and worked example are shown below. Deterministic formula executed locally in the browser.
- Assumptions
- Only entered values and stated assumptions are included. Local fees, rules and professional requirements are excluded unless explicitly shown.
- Engine version
- 1.0.0
- Production verification baseline
- Release 63 · source
998803f - Verification run
release-v63-2026-09-20- Last verified
- — Automated release checks passing
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How this calculator works
For ax²+bx+c=0 with a≠0, calculate D=b²−4ac. The roots are (−b±√D)/(2a); D>0 gives two real roots, D=0 one repeated real root, and D<0 a complex-conjugate pair. The vertex is (−b/(2a), f(−b/(2a))).
Variables and inputs
- Coefficient a
- Required numeric input. finite value from −1 trillion to 1 trillion; must be non-zero.
- Coefficient b
- Required numeric input. finite value from −1 trillion to 1 trillion.
- Coefficient c
- Required numeric input. finite value from −1 trillion to 1 trillion.
How to interpret the result
The calculator returns discriminant and root classification, two real roots, a repeated root or a complex-conjugate pair, vertex and axis of symmetry, numerical substitution residual for each root. Read each output against the assumptions you entered, compare alternatives one variable at a time, and use the result as an estimate rather than a promise.
Common mistakes to avoid
- Mixing units or entering a rate as a decimal when the field requests a percentage.
- Rounding intermediate values instead of the final displayed result.
- Leaving out a cost, timing rule or local requirement listed in the assumptions below.
Practical uses
Use the quadratic equation calculator to check a real scenario, compare a changed assumption and explain the arithmetic before relying on a quote, specification or decision.
Worked example
For x²−5x+6=0, D=(−5)²−4(1)(6)=1. The roots are (5±1)/2, so x=3 and x=2. The vertex is (2.5,−0.25), and substituting either root returns a residual of zero.
Assumptions and limitations
- coefficients represent a one-variable quadratic polynomial in standard form
- complex roots use i as the imaginary unit
- all arithmetic uses finite JavaScript numbers and results are rounded only for display
- very ill-conditioned coefficients can lose floating-point precision even though a numerically stable real-root form is used
- a=0 is rejected rather than silently solving a different linear equation
- substitution residuals are numerical checks, not symbolic proofs
Important notes
Results are estimates based on the values entered. Confirm rates, prices, units and professional requirements that apply in your location. Review the Answer Atlas Calculation Standard and corrections policy for formula, privacy and correction rules.
Frequently asked questions
What does a negative discriminant mean?
It means the quadratic has no real roots. It still has two complex conjugate roots, which this calculator reports as a real part plus or minus an imaginary part.
Why must coefficient a be non-zero?
If a is zero, the x² term disappears and the equation is linear rather than quadratic. The calculator rejects that case instead of changing methods silently.
What is the substitution residual?
The calculator substitutes each displayed root into ax²+bx+c and reports the remaining magnitude. Zero or a very small value confirms the numerical root within floating-point precision.