Algebra · Parabolas
Quadratic
Formula Calculator
Solve ax²+bx+c=0 for real and complex roots. Shows discriminant, step-by-step solution, parabola chart, vertex, axis of symmetry, and all key properties.
—
Root x₁
—
Root x₂
—
Discriminant
Enter Coefficients of ax² + bx + c = 0
1x² +
−5x +
6 = 0
quadratic
linear
constant
💡 Fractional values like 3/4 are not supported — convert to decimal (0.75) first. Coefficient a cannot be zero.
Roots of the Equation
Δ = b²−4ac = 1
x₁ = 3Root 1
x₂ = 2Root 2
2 Real Roots
Discriminant Δ
—
Vertex
—
x-intercepts
—
Opens
—
📋 Step-by-Step Solution
📈 Parabola Graph
📊 Parabola Properties
The Quadratic Formula
A quadratic equation has the form ax²+bx+c=0, where a≠0. The quadratic formula finds all solutions (roots) directly from the coefficients without factoring or completing the square.
Formula & Discriminant
x = (−b ± √(b²−4ac)) / 2a
Discriminant: Δ = b² − 4ac
Δ > 0 → 2 distinct real roots
Δ = 0 → 1 repeated real root (double root)
Δ < 0 → 2 complex conjugate roots (no real roots)
Vertex: (−b/2a, c − b²/4a)
Axis of sym: x = −b / 2a
Sum of roots: x₁ + x₂ = −b/a
Product: x₁ × x₂ = c/a
What does the discriminant tell you?
The discriminant Δ=b²−4ac determines the nature of roots without fully solving. Δ>0: two different real roots — parabola crosses x-axis twice. Δ=0: one repeated root — parabola touches x-axis at exactly one point (vertex on x-axis). Δ<0: no real roots — parabola doesn't cross the x-axis. The roots are complex conjugates a±bi.
How do I solve a quadratic by completing the square?
Move constant to right: ax²+bx = −c. Divide by a: x²+(b/a)x = −c/a. Add (b/2a)² to both sides: x²+(b/a)x+(b/2a)² = (b/2a)²−c/a. Factor left: (x+b/2a)² = (b²−4ac)/4a². Take square root of both sides. Solve for x. This is algebraically equivalent to the quadratic formula.
What are complex roots and what do they mean geometrically?
When Δ<0, the roots are complex numbers of the form x = (−b ± i√|Δ|) / 2a where i=√−1. Geometrically, this means the parabola doesn't intersect the real x-axis — it sits entirely above (a>0) or below (a<0) the x-axis. The complex roots still describe the equation algebraically.