Solve any quadratic equation ax² + bx + c = 0 instantly — find both roots, discriminant, and nature of roots using the quadratic formula.
The Quadratic Equation Solver instantly finds the roots of any quadratic equation of the form ax² + bx + c = 0 using the quadratic formula. Quadratic equations are one of the most important topics in algebra, taught from Class 9 onwards in India and appearing in virtually every mathematics examination — CBSE board exams, ICSE, JEE Main, JEE Advanced, NEET mathematics sections, and all competitive aptitude tests.
This solver computes both roots (x₁ and x₂), the discriminant (D), and tells you the nature of the roots — whether they are distinct real roots, equal roots, or complex (non-real) roots.
Understanding Quadratic Equations
A quadratic equation is a second-degree polynomial equation in one variable x. The "quadratic" in the name comes from "quadratus" (Latin for square), referring to the x² term. Every quadratic equation has exactly two roots (counting multiplicity), which may be:
- Two distinct real roots (when D > 0): The parabola crosses the x-axis at two points.
- Two equal real roots (when D = 0): The parabola just touches the x-axis at one point (a perfect square).
- Two complex/imaginary roots (when D < 0): The parabola doesn't cross the x-axis at all.
Real-World Applications
Quadratic equations appear in physics (projectile motion: how high does a ball go? when does it land?), engineering (bridge arc design), economics (profit maximization), and computer graphics (Bezier curves). Any situation involving area, acceleration, or optimization often leads to a quadratic equation.
1. Enter coefficient a: The coefficient of x² (must be non-zero — if a=0, it's a linear equation).
2. Enter coefficient b: The coefficient of x.
3. Enter coefficient c: The constant term.
4. View Results: The calculator computes the discriminant D = b²−4ac and then finds both roots using the quadratic formula. It also tells you the nature of the roots.
Quadratic Formula: x = (−b ± √(b² − 4ac)) / (2a)
Discriminant: D = b² − 4ac
- D > 0: Two distinct real roots
- D = 0: Two equal real roots (x = −b/2a)
- D < 0: No real roots (complex roots)
Two roots:
- Root 1 = (−b + √D) / (2a)
- Root 2 = (−b − √D) / (2a)
Sum of roots: x₁ + x₂ = −b/a
Product of roots: x₁ × x₂ = c/a
Example 1 — Two distinct roots: x² − 5x + 6 = 0 (a=1, b=−5, c=6)
D = 25 − 24 = 1 > 0
Root 1 = (5 + 1)/2 = 3, Root 2 = (5 − 1)/2 = 2
Verification: (x−2)(x−3) = x²−5x+6 ✓
Example 2 — Equal roots: x² − 4x + 4 = 0 (a=1, b=−4, c=4)
D = 16 − 16 = 0
Root = 4/2 = 2 (double root)
Example 3 — No real roots: x² + x + 1 = 0 (a=1, b=1, c=1)
D = 1 − 4 = −3 < 0 → No real roots