Your result
Click Calculate
Enter your values, then click Calculate result.How this calculator helps
Solve numeric quadratic equations with real or complex roots, including repeated roots, linear reductions and degenerate equations. Move all terms to one side and match their coefficients to ax²+bx+c=0. A missing term has coefficient zero, and the signs must reflect the rearranged equation. For example, x²=5x−6 becomes x²−5x+6=0. The inputs accept finite real numbers within ±10¹², not symbolic expressions or a written equation. Fractional coefficients must be entered as decimal numbers.
How to use it
- 1
Select the correct input roles for quadratic equation calculator and enter the values described below. The demonstration defaults illustrate the method; they are not independently verified personal measurements or live market data.
- 2
Review discriminant classification and the original source record. Match units, signs and the chosen mode before submitting, rather than relying on a familiar-looking default number.
- 3
Click Calculate result to submit the current fields. Editing an input preserves the prior submitted output until you calculate again; the pending-change message distinguishes that saved output from the new values.
- 4
Check the labeled output against the worked example and independent verification steps. Review verification and limits before using the result in another document, and keep the complete input basis with a copied answer.
Formula and methodology
For ax²+bx+c=0, discriminant Δ=b²−4ac. Roots are (−b±√Δ)/(2a). If a=0, inspect the remaining linear or constant equation.
The calculator applies the displayed arithmetic to the values entered on this device. It does not silently load a local tax rate, currency conversion or commercial assumption.
Worked calculation example
For a=1, b=−5 and c=6, the equation is x²−5x+6=0. Its discriminant is 25−24=1 and its roots are 3 and 2. Substituting either into the original equation gives zero. For x²+1=0, the discriminant is negative and the roots are 0+i and 0−i rather than an input error.
How to interpret your result
The implementation scales coefficients before evaluating the discriminant and uses a cancellation-resistant form for the real roots. It computes one root from a signed square-root expression and the other from the product relationship when possible. This reduces loss of the smaller root when b is much larger than the other terms. It remains floating-point arithmetic and does not guarantee exact rational or radical output.
For different inputs or formulas, use Substitution Calculator — Two Linear Equations; Eigenvalue Calculator — 2×2 Matrices; Nth Root Calculator.
Related questions this calculator covers
- quadratic calculator
Scenario comparison
| Scenario | What it shows |
|---|---|
| Repeated root | x²−2x+1 gives x=1 twice. |
| Linear reduction | 0x²+2x−6 gives x=3. |
| Constant identity | all-zero coefficients make every x a solution. |
Common mistakes to avoid
- Forgetting to reverse a sign when moving a term.
- Discarding complex roots as invalid arithmetic.
- Using the quadratic formula when a is zero.
Substitute both real roots and check the residual. Independently verify that their sum equals −b/a and product equals c/a for a nonzero a, allowing floating-point rounding. Test the complex example x²+1 and the linear and constant boundaries. Scaling all three coefficients by the same nonzero factor should preserve the solutions.
Authoritative reference. Method and scope reviewed on 5 October 2026. SolvePilot provides the original worked example and bounded browser implementation. Editorial and arithmetic review by Mohammad Qasim does not certify user measurements, a real contract or an individual professional decision. The reference supplies method, unit or source-record context; it does not approve this implementation or its inputs.What can affect the result?
Enter coefficients
Move all terms to one side and match their coefficients to ax²+bx+c=0. A missing term has coefficient zero, and the signs must reflect the rearranged equation. For example, x²=5x−6 becomes x²−5x+6=0. The inputs accept finite real numbers within ±10¹², not symbolic expressions or a written equation. Fractional coefficients must be entered as decimal numbers.
Discriminant classification
A positive discriminant gives two real roots, zero gives a repeated real root, and a negative discriminant gives a complex conjugate pair. Those classifications refer to the supplied coefficients as represented numerically. A value very close to zero can be sensitive to rounding. The worksheet does not silently round a small discriminant to zero or infer exact symbolic equality from decimal approximations.
Stable real roots
The implementation scales coefficients before evaluating the discriminant and uses a cancellation-resistant form for the real roots. It computes one root from a signed square-root expression and the other from the product relationship when possible. This reduces loss of the smaller root when b is much larger than the other terms. It remains floating-point arithmetic and does not guarantee exact rational or radical output.
Linear and constant cases
When a is exactly zero, the expression is no longer quadratic. A nonzero b gives the linear solution −c/b. If a and b are zero but c is nonzero, no x can solve the equation. If all three coefficients are zero, every x satisfies it. The page reports these cases directly instead of dividing by a zero quadratic coefficient.
Verification and limits
For real roots, substitute each value into the original polynomial and compare the residual with the scale of its terms. Complex roots can be checked through their sum and product or complex arithmetic. Display rounding means a copied decimal may leave a small residual. This page does not solve systems, inequalities, higher-degree polynomials or equations with complex coefficients; retain the coefficients and the root classification with your answer.
Privacy and browser processing
Values entered on this page are processed in the current browser session. SolvePilot does not require an account and does not receive the values entered into the calculator. Refreshing or closing the page clears the working values unless the browser itself restores a previous session. Avoid entering identifying or account information because the calculation needs summary values only.
Accuracy and verification
Accuracy depends first on input quality. Confirm definitions, scales, dates and source information before entering a value. Keep an independent record of any result used for planning because this page does not create an official statement or retain a calculation history.
Limits of this estimate
For real roots, substitute each value into the original polynomial and compare the residual with the scale of its terms. Complex roots can be checked through their sum and product or complex arithmetic. Display rounding means a copied decimal may leave a small residual. This page does not solve systems, inequalities, higher-degree polynomials or equations with complex coefficients; retain the coefficients and the root classification with your answer.
Sources and review information
Frequently asked questions
What if a is zero?＋
The tool handles the remaining linear or constant equation explicitly. A nonzero b gives one linear solution; a constant equation can have no solution or every x.
Can roots be complex?＋
Yes. A negative discriminant produces two conjugate roots with a real part and an imaginary magnitude.
Which root is first?＋
The stable computation determines the order. It is not a promised ascending order, and both labeled roots should be considered.
Can I enter an equation directly?＋
No. Rearrange it into ax²+bx+c=0 and enter its numeric coefficients in the three fields.
Why is substitution slightly different from zero?＋
Displayed decimal roots are rounded, and browser floating-point operations also have finite precision. Judge residuals relative to the original coefficient scale.