Use this calculator to find the roots (solutions) of a quadratic equation in the form ax² + bx + c = 0. It supports real and complex roots.
Equation: x² - 3x + 2 = 0
What does “finding roots” mean?
In algebra, a root of an equation is a value of x that makes the expression equal to zero. For a quadratic equation, that means solving:
ax² + bx + c = 0
If you graph this equation as a parabola, the roots are the x-values where the curve intersects the x-axis.
How this calculator works
This root finder uses the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
The term inside the square root, b² - 4ac, is called the discriminant. The discriminant tells you what kind of roots you have:
- Discriminant > 0: Two distinct real roots
- Discriminant = 0: One repeated real root
- Discriminant < 0: Two complex conjugate roots
Quick examples
Example 1: Two real roots
Equation: x² - 3x + 2 = 0
Roots: x = 1 and x = 2
Example 2: One repeated root
Equation: x² - 4x + 4 = 0
Root: x = 2 (double root)
Example 3: Complex roots
Equation: x² + 2x + 5 = 0
Roots: x = -1 + 2i and x = -1 - 2i
Why roots matter
Root-finding appears in many fields, including:
- Physics: trajectory and motion equations
- Engineering: control systems and signal analysis
- Finance: break-even models and optimization problems
- Computer science: numerical methods and modeling
Tips for accurate input
- Enter all three coefficients: a, b, and c.
- If a term is missing, use 0 for that coefficient.
- If a = 0, the equation becomes linear, and the tool will solve it as bx + c = 0.
Final note
This calculator is designed for fast checks, homework support, and concept review. If you need advanced equation solving (higher-degree polynomials, systems of equations, or symbolic factoring), you can pair this with graphing or CAS tools for deeper analysis.