Quadratic Equation Solver — ax² + bx + c = 0
Solve any quadratic equation with step-by-step working. Find the discriminant, both roots (real or complex), vertex, and axis of symmetry. 100% private — runs in your browser.
How to Solve Quadratic Equations
A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b, and c are real numbers and a is not zero. The quadratic formula provides a universal method to find the values of x (called roots or solutions) that satisfy the equation. This calculator applies the formula automatically and shows every step so you can learn the process or verify your homework.
The quadratic formula is one of the most important formulas in algebra. It is used in physics (projectile motion), engineering (signal processing), economics (profit maximization), and countless other fields. Understanding how to interpret the discriminant and roots is a foundational math skill.
The Quadratic Formula
x = (-b ± √(b² - 4ac)) / 2a
Discriminant: Δ = b² - 4ac
Vertex: (-b/2a, c - b²/4a)
Where:
- a = coefficient of x² (must not be zero)
- b = coefficient of x
- c = constant term
- Δ > 0: two distinct real roots
- Δ = 0: one repeated real root
- Δ < 0: two complex conjugate roots
Example: Solve 2x² + 5x - 3 = 0
- a = 2, b = 5, c = -3
- Discriminant = 5² - 4(2)(-3) = 25 + 24 = 49
- √49 = 7
- x₁ = (-5 + 7) / (2 × 2) = 2/4 = 0.5
- x₂ = (-5 - 7) / (2 × 2) = -12/4 = -3
Understanding the Discriminant
The discriminant (Δ = b² - 4ac) is the key to understanding what type of roots a quadratic equation has before you even solve it. A positive discriminant means the parabola crosses the x-axis at two points (two real roots). A zero discriminant means the parabola touches the x-axis at exactly one point (a repeated root, also called a double root). A negative discriminant means the parabola never crosses the x-axis (two complex conjugate roots involving the imaginary unit i).
Vertex Form and Axis of Symmetry
Every quadratic equation y = ax² + bx + c can be rewritten in vertex form y = a(x - h)² + k, where (h, k) is the vertex of the parabola. The axis of symmetry is the vertical line x = h = -b/(2a). If a is positive, the parabola opens upward and the vertex is the minimum point. If a is negative, the parabola opens downward and the vertex is the maximum point. This information is essential for graphing and optimization problems.
Applications of Quadratic Equations
Quadratic equations appear everywhere: projectile motion in physics (height = -½gt² + v₀t + h₀), profit and revenue optimization in business, area and dimension problems in geometry, circuit analysis in electrical engineering, and population growth models in biology. Mastering the quadratic formula gives you a powerful tool for solving real-world problems across many disciplines.