Solve ax² + bx + c = 0 using the quadratic formula. Real and complex roots.
A quadratic equation is a second-degree polynomial equation of the form ax² + bx + c = 0, where a ≠ 0. Quadratic equations appear in projectile motion, area calculations, profit maximization, and many engineering problems.
The Quadratic Equation Solver on Calculator Expert applies the quadratic formula to find all roots of any quadratic equation. It handles real roots (two distinct, one repeated) and complex roots when the discriminant is negative, showing full working steps.
| Discriminant | Root Type | Graph Behavior |
|---|---|---|
| D > 0 | Two distinct real roots | Parabola crosses x-axis twice |
| D = 0 | One repeated real root | Parabola touches x-axis |
| D < 0 | Two complex roots | Parabola doesn't cross x-axis |
Enter your values in the empty input fields above and click "Calculate." All fields start empty so you can input any values you need. The result is displayed instantly with the working formula. Calculator Expert provides accurate, ad-free calculations for students, teachers, and professionals.
Method 1: Quadratic Formula: Directly substitute a, b, c into x = (−b ± √(b²−4ac)) / 2a. Calculate the discriminant first to determine the nature of the roots.
Method 2: Factoring Method: If the quadratic can be factored as (px+q)(rx+s)=0, then roots are x = −q/p and x = −s/r. This method is faster when applicable but the quadratic formula always works.
The coefficient 'a' cannot be zero (that would make it a linear equation). Very large coefficients may cause floating point overflow. Complex roots are expressed in a+bi form with real and imaginary parts rounded to 6 decimal places.
Some health index models use quadratic relationships between body dimensions. The quadratic formula is a fundamental algebraic tool that appears in growth modeling and biometric scaling equations used in nutritional science.
Quadratics are used in calculating optimal pricing for maximum profit, projectile trajectory in physics, signal processing filters, engineering beam deflection, and geometric area problems.