Quadratic Equation Solver
Enter coefficients a, b, and c to solve ax² + bx + c = 0. Get roots, discriminant, vertex, and full step-by-step working instantly.
Solution Results
Step-by-Step Working
How to Use the Quadratic Equation Solver
Enter Coefficient a — Type the value of 'a' (the x² term). This cannot be zero, as that would make the equation linear, not quadratic.
Enter Coefficient b — Type the value of 'b' (the x term). Can be zero or negative — for example: −5, 0, 3.5.
Enter Constant c — Type the constant 'c'. Can be any real number including zero or a decimal.
Click Solve — Press the Solve Equation button to calculate roots, discriminant, and vertex. You can also press Enter from any field.
Review Results — Read the roots (real or complex), discriminant, vertex coordinates, and step-by-step working displayed below.
Key Features of This Tool
Real & Complex Roots
Handles all discriminant cases — positive, zero, and negative — returning complex roots in p ± qi format.
Vertex Coordinates
Instantly shows the vertex (h, k) of the parabola — essential for graphing and optimization problems.
Step-by-Step Working
Displays full working using the quadratic formula — ideal for exam preparation and concept verification.
Decimal Coefficients
Accepts integers, decimals, and negatives for all three coefficients without rounding errors.
Vieta's Formulas
Shows sum and product of roots using Vieta's relations — useful for verifying solutions quickly.
Mobile Friendly
Fully responsive layout that works perfectly on phones, tablets, and desktops — no app needed.
Formula and How It Works
The quadratic formula is the universal method for solving any equation of the form ax² + bx + c = 0 where a ≠ 0. It derives directly from the process of completing the square.
This gives two roots: x₁ using (+) and x₂ using (−). The expression b² − 4ac is called the discriminant.
The discriminant D = b² − 4ac determines the nature of roots:
- D > 0 — Two distinct real roots (parabola crosses x-axis at two points)
- D = 0 — One repeated real root (parabola is tangent to x-axis)
- D < 0 — Two complex conjugate roots (parabola does not touch x-axis)
The vertex of the parabola gives the minimum (if a > 0) or maximum (if a < 0) point:
Also written as (h, k) where h = −b/(2a) and k = f(h)
| Variable | Name | Role |
|---|---|---|
| a | Leading coefficient | Controls parabola width and direction |
| b | Linear coefficient | Shifts the axis of symmetry |
| c | Constant term | y-intercept of the parabola |
| D | Discriminant | b²−4ac; determines root nature |
| x₁, x₂ | Roots / Zeros | x-values where parabola meets x-axis |
By Vieta's formulas: x₁ + x₂ = −b/a and x₁ × x₂ = c/a. These let you verify roots without back-substitution.
Practical Examples
Priya — Mumbai, India
Class 10 student solving x² − 5x + 6 = 0 for her board exam. Coefficients: a=1, b=−5, c=6.
x = (5 ± √1) / 2
x₁ = (5+1)/2 = 3 | x₂ = (5−1)/2 = 2
Rajan — Bengaluru, India
JEE aspirant checking whether 4x² − 4x + 1 = 0 has equal roots. Coefficients: a=4, b=−4, c=1.
x = (4 ± 0) / 8 = 0.5
Lukas — Berlin, Germany
Engineering student solving x² + 2x + 5 = 0. Expects complex roots. Coefficients: a=1, b=2, c=5.
x = (−2 ± √(−16)) / 2 = −1 ± 2i
Meena — Chennai, India
Finding the vertex of 2x² − 8x + 3 = 0 for a physics projectile problem. a=2, b=−8, c=3.
D = 64 − 24 = 40
x = (8 ± √40) / 4
What Is a Quadratic Equation?
A quadratic equation is a second-degree polynomial equation in a single variable x, written in the standard form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. The word "quadratic" comes from the Latin quadratus, meaning square — a reference to the x² term that defines it.
Quadratic equations appear across mathematics, science, and engineering. They describe the trajectory of a ball thrown in the air, the shape of satellite dishes and telescope lenses, break-even analysis in business, and the curvature of roads and bridges. Every parabola in nature can be described by a quadratic equation.
Solving a quadratic equation means finding the values of x — called roots, zeros, or solutions — that make the equation true. The quadratic formula method works for every quadratic, regardless of whether real or complex roots exist, making it the most reliable approach for any exam or practical problem.
Quadratic Equation in Different Languages
Want a deeper understanding of quadratic equations — including history, derivation, and real-world applications? Read our full guide.
Read the Complete Guide →Frequently Asked Questions
Is this tool free to use?
What is a quadratic equation?
What does the discriminant tell us?
Can this solver handle complex roots?
What happens if I enter a = 0?
What is the vertex of a parabola?
Can I enter decimal or negative coefficients?
What is the quadratic formula?
How is the sum and product of roots calculated?
Does this tool work on mobile?
Can I use this for JEE, NEET, or board exam practice?
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