Quadratic Equation Solver
Solve any quadratic equation ax² + bx + c = 0 using the quadratic formula. Finds real and complex roots.
Did this tool work for you?
How to use this calculator
The discriminant b²−4ac determines the nature of roots: positive = two real roots, zero = one repeated root, negative = two complex roots.
- 1
Enter the coefficient a (of x²), b (of x), and c (constant).
- 2
For example, x² − 5x + 6 = 0 uses a=1, b=−5, c=6.
- 3
The calculator applies the quadratic formula and shows roots.
- 4
A negative discriminant means the roots are complex (imaginary).
Frequently asked questions
What if the discriminant is negative?
The equation has no real solutions. The two roots are complex conjugates in the form p ± qi, where i = √−1. Complex roots appear in pairs when the parabola does not cross the x-axis.
How do I verify my roots are correct?
Substitute each root back into the equation. If ax² + bx + c = 0 is satisfied, the root is correct. Alternatively, the product of the roots = c/a and the sum of the roots = −b/a (Vieta's formulas).
Can I use this for equations not in standard form?
Rearrange first. For example, 2x² = 3x − 1 becomes 2x² − 3x + 1 = 0, so a=2, b=−3, c=1. Always move all terms to one side before identifying a, b, and c.
What is the vertex of the parabola?
The vertex x-coordinate is −b/(2a). Substitute back to get the y-coordinate. The vertex is the minimum (if a > 0) or maximum (if a < 0) point of the parabola y = ax² + bx + c.
Solving quadratic equations step by step
How to use the quadratic equation
Use this quadratic equation to olve any quadratic equation ax² + bx + c = 0 using the quadratic formula. Enter your values above and get your result in seconds. The tool is free, works on all devices, and keeps your data private — nothing is stored or shared.
How the quadratic equation works
The quadratic equation solver uses standard formulas used in mathematics, science, and engineering. Enter your inputs, and the tool calculates the result instantly in your browser. No server-side processing means your data stays on your device. Results update in real time as you change inputs.
The quadratic formula
The quadratic formula x = (−b ± √(b²−4ac)) / 2a solves any equation of the form ax² + bx + c = 0. It was derived by completing the square and works universally — even when factoring is not obvious.
Three methods to solve quadratic equations
Factoring: fastest when the roots are rational integers. Completing the square: works always, useful for deriving the vertex form. Quadratic formula: the universal method. For exams, the quadratic formula is the most reliable when factoring is not obvious.
Real-world applications of quadratic equations
Quadratics model projectile motion (height vs time), area problems (find dimensions given area), break-even analysis (where revenue equals cost), and parabolic structures in engineering and architecture. Any U-shaped relationship between two quantities is likely quadratic.
Quadratic equation: how it works
Built on well-established mathematical principles, this tool delivers accurate results for students, researchers, and professionals. Enter your values and get instant clarity without specialist software or manual arithmetic.
Who uses this tool?
Teachers, students, engineers, and data analysts use it to verify calculations, check homework, and solve problems faster. It is intuitive enough for beginners while comprehensive enough for professional use.
Learn more from an authoritative source:
WikipediaScientific Calculator
Evaluate mathematical expressions including trigonometry, logarithms, exponents, and more.
Percentage Calculator
Quickly calculate percentages, percentage change, and percentage of a total.
Fraction Calculator
Add, subtract, multiply, or divide two fractions and get the simplified result instantly.
Ratio Calculator
Simplify ratios, find missing values, and scale ratios up or down.
Results are estimates for informational purposes only and do not constitute professional financial, medical, legal, or technical advice. Read full disclaimer →