Quadratic Equation Solver
Solve any quadratic equation ax² + bx + c = 0. Find real or complex roots, discriminant, and vertex.
Examples
Two real roots (2 and 3)
Discriminant
1
Root x₁
3
Root x₂
2
Vertex X
2.5
Vertex Y
-0.25
Two distinct real roots: x₁ = 3 and x₂ = 2.
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Examples
How It Works
Formula
Variables
- Coefficient of x² (must be non-zero)
- Coefficient of x
- Constant term
The discriminant b²−4ac is computed first. If non-negative, real roots are found with the quadratic formula. If negative, complex roots are expressed as a±bi. The vertex is at x = −b/(2a).
Frequently Asked Questions
01What is a quadratic equation?
A quadratic equation is a polynomial equation of degree 2 in the form ax² + bx + c = 0, where a ≠ 0.
02What is the discriminant?
The discriminant Δ = b² − 4ac determines the nature of the roots: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated root, Δ < 0 gives two complex conjugate roots.
03What is the quadratic formula?
x = (−b ± √(b² − 4ac)) / (2a). It gives the solutions of any quadratic equation.
04What are complex roots?
When the discriminant is negative, the roots contain imaginary numbers (i = √−1), expressed as a ± bi.
05What is the vertex of a parabola?
The vertex is the highest or lowest point. Its x-coordinate is −b/(2a) and y-coordinate is f(−b/(2a)).