Quadratic Equation Solver
Enter coefficients for the equation ax² + bx + c = 0.
What is a quadratic equation?
A quadratic equation is a second-degree polynomial equation written in the standard form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. Because the highest power is 2, its graph is always a parabola.
Quadratic equations appear everywhere: projectile motion in physics, optimization in economics, shape design in engineering, and even computer graphics. A fast quadratic calculator helps you solve these equations without manual algebra errors.
How this quadratic calculator works
This calculator uses the classic quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
The expression under the square root, b² - 4ac, is called the discriminant. It determines the type of roots you get.
Discriminant interpretation
- D > 0: two distinct real roots
- D = 0: one repeated real root
- D < 0: two complex conjugate roots
Additional values provided
Beyond roots, this solver also reports:
- Vertex of the parabola: the turning point
- Axis of symmetry: vertical line through the vertex
- Equation preview: your exact equation formatting
Example problems
Example 1: x² - 3x + 2 = 0
Set a = 1, b = -3, c = 2. The roots are x = 1 and x = 2.
Example 2: 2x² + 4x + 2 = 0
Set a = 2, b = 4, c = 2. The discriminant is zero, so there is one repeated root: x = -1.
Example 3: x² + 2x + 5 = 0
Set a = 1, b = 2, c = 5. The discriminant is negative, producing complex roots: x = -1 ± 2i.
Common mistakes to avoid
- Forgetting that a cannot be zero for a true quadratic equation
- Dropping parentheses around -b in the formula
- Mixing up signs when calculating b² - 4ac
- Assuming roots are always real numbers
Why this tool is useful
Whether you are a student checking homework, a teacher preparing examples, or a professional doing quick modeling, this quadratic calculator gives immediate, reliable output. It reduces arithmetic friction and keeps your focus on interpretation and decision-making.