Solve ax² + bx + c = 0
Enter the coefficients below and click Calculate Roots. This tool handles real roots, repeated roots, and complex roots.
What is a quadratic equation?
A quadratic equation is a second-degree polynomial equation in one variable, usually written as:
ax² + bx + c = 0
where a, b, and c are constants and a ≠ 0. The solutions to this equation are called the roots (or zeros) of the quadratic.
How the roots are calculated
The standard method is the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
The expression inside the square root, b² - 4ac, is called the discriminant. It tells you what type of roots to expect.
Root types based on discriminant
- Discriminant > 0: two distinct real roots
- Discriminant = 0: one repeated real root
- Discriminant < 0: two complex conjugate roots
Why this calculator is useful
Quadratic equations appear constantly in math, physics, economics, engineering, and computer science. A quick calculator helps you:
- check homework answers,
- validate algebra steps,
- analyze parabolic models, and
- save time during exam prep.
Example problems
Example 1: Two real roots
For x² - 5x + 6 = 0, we have a=1, b=-5, c=6. The discriminant is 1, so there are two real roots: x=2 and x=3.
Example 2: Repeated root
For x² - 4x + 4 = 0, the discriminant is 0. The equation has a repeated root at x=2.
Example 3: Complex roots
For x² + 2x + 5 = 0, the discriminant is -16. The roots are complex: x = -1 ± 2i.
Common mistakes to avoid
- Forgetting that a cannot be zero in a quadratic equation.
- Sign errors when substituting negative values of b or c.
- Incorrectly simplifying square roots, especially with negative discriminants.
- Ignoring that complex roots come in conjugate pairs.
Practical applications
Quadratics are used to model projectile motion, revenue optimization, area and geometry constraints, signal processing, and many machine-learning loss curves. Understanding roots helps identify when outcomes cross zero, when turning points occur, and whether solutions are physically meaningful.
Final takeaway
This quadratic equation roots calculator gives fast, reliable output along with the discriminant and interpretation of root type. Use it as a learning companion, not just a shortcut—if you understand the structure of ax² + bx + c = 0, you can solve a huge class of real-world problems more confidently.