Quadratic Equation Solver
Find the roots of any quadratic equation of the form ax² + bx + c = 0
Equation Coefficients
Enter the coefficients a, b, and c for ax² + bx + c = 0
Solution
Quadratic Equation Solver: Find Roots Instantly
Welcome to the Quadratic Equation Solver, a powerful algebra calculator designed to find the exact roots of any second-degree polynomial equation. Whether you are solving for x-intercepts in a high school math class or computing parabolic trajectories in physics, this tool completely automates the Quadratic Formula. By simply inputting your a, b, and c coefficients, our engine evaluates the discriminant and traces out the exact step-by-step arithmetic to reach your real or complex roots.
The Quadratic Formula
Standard Form: ax² + bx + c = 0
The Fundamental Theorem of Algebra guarantees that every quadratic equation has exactly two roots. These roots can be two unique real numbers, one repeated real number (a double root), or a conjugate pair of complex numbers involving i.
Understanding the Discriminant (Δ)
The expression tucked inside the square root of the formula, b² - 4ac, is known as the Discriminant (denoted by Δ). It acts as an indicator, instantly revealing the specific nature of the roots without having to solve the entire equation:
- Δ > 0 (Positive)
Two distinct real roots. The parabola intercepts the x-axis at two completely different coordinates.
- Δ = 0 (Zero)
One real root. The parabola perfectly grazes the x-axis with its vertex, creating a single repeated solution.
- Δ < 0 (Negative)
Two complex roots. The parabola never touches the x-axis, resulting in imaginary numbers (i = √-1).
Example Calculation
Solving: 2x² - 4x - 6 = 0
Coefficients: a = 2, b = -4, c = -6
- Compute the Discriminant: Δ = (-4)² - 4(2)(-6) = 16 - (-48) = 64.
- Since Δ is positive, expect two real roots. √64 = 8.
- Root 1: ( -(-4) + 8 ) / (2*2) = (4 + 8) / 4 = 12 / 4 = 3
- Root 2: ( -(-4) - 8 ) / (2*2) = (4 - 8) / 4 = -4 / 4 = -1
Result: x = 3, x = -1
Academic & Scientific References
For further understanding and validation of the formulas used above, we recommend exploring these authoritative resources: