Quadratic Equation Calculator (Solve ax² + bx + c = 0)
Find all roots, the discriminant and the vertex of any quadratic ax² + bx + c = 0.
Enter coefficients of ax² + bx + c = 0
Result
Root x₁
3
Root x₂
2
Discriminant (Δ)
1
Nature
Two distinct real roots
Vertex
(2.5, -0.25)
Sum of roots
5
= −b/a
Product of roots
6
= c/a
Concept
A quadratic is any polynomial of degree 2. Its graph is a parabola — opens upward if a > 0, downward if a < 0.
The discriminant Δ = b² − 4ac tells you the nature of roots: Δ > 0 two real roots, Δ = 0 one repeated root, Δ < 0 two complex conjugate roots.
The quadratic formula is derived by completing the square — a technique that rewrites ax² + bx + c as a(x + b/2a)² + (c − b²/4a).
Formula
x=−b ± √b2 − 4ac2a
Discriminant Δ=b2 − 4ac
Vertex=xv = −b ÷ 2a, yv = c − b2 ÷ 4a
Variables
a- Coefficient of x², must be non-zero.
b- Coefficient of x.
c- Constant term.
Δ- Discriminant = b² − 4ac.
x_v- x-coordinate of parabola vertex = −b/2a.
y_v- y-coordinate of vertex = c − b²/4a.