Quadratic Equation Solver

Input Coefficients

Solution

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of the second degree, written in the standard form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

Quadratic equations appear in many real-world applications including physics (projectile motion), engineering, economics, and geometry. Solutions to these equations are called roots or zeros.

How to Use This Solver

1. Enter the coefficients: Input values for a, b, and c in the input fields above.
2. Click Solve: Press the “Solve Equation” button to calculate the roots and analysis.
3. Review the solution: The tool displays the discriminant, root type, vertex coordinates, exact roots, and step-by-step working.
4. View the graph: A visual representation of the parabola is drawn with roots marked on the x-axis.
5. Copy results: Click on any output section to copy to clipboard.

Common Use Cases

• Physics: Solving projectile motion problems and calculating trajectory paths.
• Engineering: Designing parabolic structures, antennas, and reflectors.
• Finance: Modeling profit/loss functions and break-even analysis.
• Education: Learning algebra and understanding quadratic functions in high school mathematics.
• Optimization: Finding minimum or maximum values in various applications.

Frequently Asked Questions

What is the discriminant?

The discriminant (Δ) is b² – 4ac. It determines the nature of the roots: if Δ > 0, there are two distinct real roots; if Δ = 0, there is one repeated real root; if Δ < 0, there are two complex conjugate roots.

What is the quadratic formula?

The quadratic formula is x = (-b ± √Δ) / (2a), where Δ = b² – 4ac. This formula always gives the exact solutions to any quadratic equation.

What is the vertex of a parabola?

The vertex is the highest or lowest point on the parabola. For a quadratic function f(x) = ax² + bx + c, the vertex is at h = -b/(2a) and k = f(h).

Can this solver handle complex roots?

Yes! When the discriminant is negative, the roots are complex numbers in the form a + bi. The solver displays both the real and imaginary parts clearly.

What is the axis of symmetry?

The axis of symmetry is a vertical line that divides the parabola into two mirror images. Its equation is x = -b/(2a), which is the same as the x-coordinate of the vertex.

Why must coefficient a be non-zero?

If a = 0, the equation becomes bx + c = 0, which is linear, not quadratic. The quadratic formula and discriminant are only valid when the highest degree term (x²) has a non-zero coefficient.