Quadratic Equation Solver
Solve any quadratic equation ax² + bx + c = 0 instantly — find both roots, discriminant, and nature of roots using the quadratic formula.
What is the Quadratic Equation Solver?
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.
How Does the Quadratic Equation Solver Work?
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.
Formula & Calculation Method
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 Calculation
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