Enter coefficients a, b, and c to solve ax² + bx + c = 0 with the quadratic formula. Get both roots, the discriminant, and the root type instantly.
Quick Facts
Quadratic Formula
x = (−b ± √(b² − 4ac)) / (2a)
Valid for any a ≠ 0. The sign of the discriminant b² − 4ac determines whether the roots are real or complex.
Results
Calculated
Root x₁
—
First solution to ax² + bx + c = 0
Root x₂
—
Second solution to ax² + bx + c = 0
Discriminant (Δ)
—
b² − 4ac
Nature of Roots
—
Based on the discriminant's sign
Add this calculator to your website
How to use this calculator
Enter the three coefficients of your quadratic equation ax² + bx + c = 0, then click Calculate. The calculator applies the quadratic formula, x = (−b ± √(b² − 4ac)) / (2a), and returns both roots along with the discriminant and what kind of roots they are. Click Clear to reset the fields to the example equation and start a new calculation.
Understanding the inputs
a is the coefficient of x², b is the coefficient of x, and c is the constant term. All three can be any real number (positive, negative, or a decimal), except a, which cannot be 0 — if a is 0 the equation is linear (bx + c = 0), not quadratic, and the quadratic formula does not apply. The hint icons (?) provide a quick reminder of what each field means.
Interpreting the results
The two highlighted cards show Root x₁ and Root x₂, the values of x that satisfy the equation. The Discriminant (Δ) card shows b² − 4ac, the value under the square root in the formula. The Nature of Roots card tells you what that discriminant means: a positive discriminant gives two distinct real roots, zero gives one repeated real root, and a negative discriminant gives two complex conjugate roots (shown in the form p + qi and p − qi).
Frequently Asked Questions
What is the quadratic formula?
For ax² + bx + c = 0 (with a not equal to 0), the quadratic formula gives x = (−b ± √(b² − 4ac)) / (2a). It works for every quadratic equation, including ones with no real solutions.
What does the discriminant tell you?
The discriminant is b² − 4ac, the value under the square root in the quadratic formula. If it's positive, the equation has two distinct real roots. If it's exactly zero, there's one repeated real root. If it's negative, the two roots are complex conjugates rather than real numbers.
What if coefficient a is 0?
If a is 0 the equation becomes bx + c = 0, which is linear, not quadratic, and the quadratic formula no longer applies since it requires dividing by 2a. This calculator requires a nonzero value for a; solve a linear equation separately if a is 0.
Can this calculator handle complex (imaginary) roots?
Yes. When the discriminant is negative, the calculator reports both roots in the form p + qi and p − qi, where p is the real part and q is the imaginary part, rather than showing an error.
Practical Guide for Quadratic Equation Calculator - Solve ax² + bx + c = 0
Quadratic Equation Calculator - Solve ax² + bx + c = 0 is most useful when the inputs reflect the situation you are actually planning around, not a best-case estimate. Treat the result as a decision aid: it gives you a structured way to compare assumptions, spot outliers, and decide what to verify next. For Math work, the most important review lens is formula choice, units, rounding, weighting, and the exact meaning of each input.
Start with a baseline run using values you can defend. Then change one assumption at a time and watch which output moves the most. If one input dominates the result, spend your verification time there first. If several inputs have similar influence, use a conservative scenario and an optimistic scenario to create a practical range instead of relying on a single exact number.
Before acting on the result, verify the result with a manual calculation or a second method when the output affects grades, budgets, or engineering work. This is especially important when the calculator supports a purchase, project plan, performance target, or operational decision. The calculator can make the math consistent, but the quality of the conclusion still depends on current data, clear units, and assumptions that match your real constraints.
When the output looks surprising, slow down and inspect each input in order. A small change in one high-leverage field can move the final number more than several low-leverage fields combined. For Quadratic Equation Calculator - Solve ax² + bx + c = 0, that means you should first confirm the value with the greatest scale, then confirm the value with the greatest uncertainty, then rerun the calculator with conservative and optimistic assumptions. This sequence turns the calculator from a single answer into a practical decision range.
Review Checklist
Confirm every input uses the unit and time period requested by the calculator.
Run a low, expected, and high scenario so the answer has a useful range.
Check whether rounding or a missing decimal place changes the decision.
Update the calculation after each new value is known or whenever the formula structure changes.