Quadratic Equation Solver

How to Use the Quadratic Equation Solver

1. Enter coefficients: Input the values for a, b, and c in the quadratic equation ax² + bx + c = 0. The equation display updates automatically to show your input.
2. Solve the equation: Click "Solve Equation" to calculate the roots using the quadratic formula. The solver handles all cases including real, repeated, and complex roots.
3. Understand the discriminant: The discriminant (Δ = b² - 4ac) determines the nature of the roots. Positive values give two distinct real roots, zero gives one repeated root, and negative values give complex conjugate roots.
4. Review solutions: The calculator displays all roots with exact values when possible, and decimal approximations for clarity.
5. Follow the steps: The step-by-step solution shows how the quadratic formula is applied, helping you understand the solving process and verify manual calculations.

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree two in one variable, written in the standard form ax² + bx + c = 0, where a, b, and c are constants (with a ≠ 0), and x represents an unknown variable. The graph of a quadratic equation is a parabola that opens upward when a > 0 and downward when a < 0.

Quadratic equations appear throughout mathematics, physics, engineering, and economics. They model parabolic motion in physics, optimization problems in calculus, area and perimeter problems in geometry, and profit maximization in business. Understanding how to solve quadratic equations is fundamental to algebra and essential for advanced mathematics.

The Quadratic Formula

The quadratic formula provides a direct method to find the roots of any quadratic equation: x = (-b ± √(b² - 4ac)) / (2a). This formula works for all quadratic equations and is derived by completing the square on the general form ax² + bx + c = 0.

The ± symbol indicates that there are typically two solutions: one using addition and one using subtraction. The expression under the square root (b² - 4ac) is called the discriminant, and it determines whether the roots are real or complex numbers.

Understanding the Discriminant

Positive discriminant (Δ > 0): The equation has two distinct real roots. The parabola crosses the x-axis at two points. Example: x² - 5x + 6 = 0 has roots x = 2 and x = 3.

Zero discriminant (Δ = 0): The equation has one repeated real root (multiplicity 2). The parabola touches the x-axis at exactly one point, called the vertex. Example: x² - 4x + 4 = 0 has a double root at x = 2.

Negative discriminant (Δ < 0): The equation has two complex conjugate roots. The parabola does not intersect the x-axis. Example: x² + x + 1 = 0 has complex roots involving the imaginary unit i.

Applications of Quadratic Equations

Quadratic equations model projectile motion in physics, where the height of an object follows a parabolic path. In business, they help find maximum profit or minimum cost points. Engineers use them to design parabolic antennas and suspension bridges. In computer graphics, quadratic Bezier curves create smooth animations and shapes.

Students encounter quadratic equations in algebra courses and standardized tests. This solver helps verify homework answers, understand the relationship between coefficients and roots, and prepare for exams by providing instant feedback and detailed solutions.

For higher-degree equations, use the Polynomial Solver. The Square Root Calculator helps compute the discriminant. For calculus operations, try the Derivative Calculator.