Quadratic Formula CalculatorFree online math calculator — instant results, no sign-up
Solve ax² + bx + c = 0.
About the Quadratic Formula Calculator
This calculator solves quadratic equations in standard form, ax² + bx + c = 0, using the quadratic formula. It calculates the discriminant and returns both real or complex roots (x₁ and x₂). Students, educators, and engineers use it to quickly evaluate second-degree polynomial solutions.
- a
- Quadratic coefficient (multiplying x²); must not be 0 for a valid quadratic equation.
- b
- Linear coefficient (multiplying x); any real number.
- c
- Constant term; any real number.
Formula and method
D = b² − 4ac When D ≥ 0: x₁ = (−b + √D) / (2a) x₂ = (−b − √D) / (2a) When D < 0: x = (−b / (2a)) ± (√(−D) / (2a))i
- aCoefficient of the squared term x²
- bCoefficient of the linear term x
- cConstant term
- DDiscriminant (determines the nature and number of roots)
- iImaginary unit, defined as √(-1)
- x₁, x₂First and second roots of the quadratic equation
The tool first computes the discriminant D = b² − 4ac. If D is greater than or equal to zero, it computes real roots using standard square root operations; if D is negative, it splits the solution into real (-b / 2a) and imaginary (√(-D) / 2a) components. Results are rounded to six decimal places.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- a1
- b-3
- c2
Results
- x₁2
- x₂1
- Discriminant1
How to use the Quadratic Formula Calculator
- 1Enter the numeric values for coefficients a, b, and constant c.
- 2Calculate the discriminant using the formula D = b² − 4ac.
- 3Evaluate the roots via the quadratic equation according to whether D is non-negative or negative.
- 4Read the values for x₁, x₂, and the discriminant displayed rounded to six decimal places.
Reading your result
A positive discriminant (D > 0) indicates two distinct real roots, while D = 0 indicates a single repeated real root. If the discriminant is negative (D < 0), the equation has two complex conjugate roots written in the form re ± im*i.
Limitations
The calculator requires a non-zero value for coefficient a, as division by 2a is undefined when a = 0. It solves single-variable polynomial equations of degree 2 only and does not compute higher-order polynomial roots or accept complex-valued inputs.
Common questions
What happens if a is set to 0?+
If a = 0, the equation becomes linear (bx + c = 0) rather than quadratic, leading to division by zero in the formula.
What does an 'i' in the solution mean?+
The symbol 'i' denotes the imaginary unit equal to the square root of -1, indicating the roots are complex conjugate numbers because the discriminant is negative.
Why are x₁ and x₂ identical when the discriminant is zero?+
When D = 0, adding or subtracting √0 yields the same numerator -b, producing a single unique root with a multiplicity of two.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
Further reading
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Results are estimates for informational purposes and are not financial, medical or legal advice.