What is the quadratic formula?

The quadratic formula is x = (−b ± √(b²−4ac)) / (2a), used to solve any equation of the form ax²+bx+c=0.

What does the discriminant tell you?

The discriminant (b²−4ac) tells you the nature of the roots. If positive: two distinct real roots. If zero: one repeated root. If negative: two complex roots with no real solutions.

What is the vertex of a parabola?

The vertex is the turning point of the parabola at coordinates (−b/2a, f(−b/2a)). It is the minimum point when a > 0 and maximum when a < 0.

How do you factor a quadratic equation?

If the roots are x₁ and x₂, the factored form is a(x − x₁)(x − x₂). You can find the roots using the quadratic formula first, then write the factored form.

What is the axis of symmetry?

The axis of symmetry is the vertical line x = −b/(2a) that passes through the vertex. The parabola is mirror-symmetric about this line.

Quadratic Equation Solver

Solve ax²+bx+c=0 instantly. Get real or complex roots, discriminant, vertex coordinates, and factored form. Three tiers including parabola graph and cubic equation solver.

⚡ Quick Calculator Get a fast estimate

Coefficient a (ax²)

Coefficient b (bx)

Coefficient c (constant)

Root x₁

3.000000

Root x₂

2.000000

Discriminant

1.0000

Vertex

(2.5000, -0.2500)

Nature

Two distinct real roots

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How to Use the Quadratic Equation Calculator

Enter the three coefficients: a (the x² term), b (the x term), and c (the constant). The calculator applies the quadratic formula instantly.

The Extended Calculator below adds a parabola SVG graph and vertex form. The Professional Calculator solves cubic equations and 2×2 systems.

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a (coefficient of x²)

b (coefficient of x)

c (constant)

Equation: 1x² -5x +6 = 0

Two Distinct Real Roots

x₁ = 3, x₂ = 2

Discriminant

Vertex

(2.5, -0.25)

Axis of Symmetry

x = 2.5

The Quadratic Formula

x = (−b ± √(b²−4ac)) / (2a) Discriminant Δ = b²−4ac Δ > 0 → two distinct real roots Δ = 0 → one repeated real root Δ < 0 → two complex conjugate roots Vertex: (−b/2a, c − b²/4a) Axis of symmetry: x = −b/2a

Worked Example

Solve: 2x² − 7x + 3 = 0

a=2, b=−7, c=3

Discriminant: (−7)² − 4(2)(3) = 49 − 24 = 25

x = (7 ± √25) / 4 = (7 ± 5) / 4

x₁ = 12/4 = 3

x₂ = 2/4 = 0.5

Factored form: 2(x − 3)(x − 0.5)

Forms of a Quadratic Equation

Form	Expression	Useful For
Standard	ax² + bx + c	Applying the formula
Factored	a(x − x₁)(x − x₂)	Finding roots directly
Vertex	a(x − h)² + k	Identifying the turning point

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ax³ + bx² + cx + d = 0

a (coefficient of x³)

b (coefficient of x²)

c (coefficient of x)

d (constant)

Equation: 1x³ -6x² +11x -6 = 0

Cubic Roots

x₁ = 3

x₂ = 2

x₃ = 1

Cardano's Method: The cubic is reduced to depressed form t³ + pt + q = 0 via substitution, then solved analytically. The discriminant Δ = −4p³ − 27q² determines root nature.

Frequently Asked Questions

The quadratic formula is x = (−b ± √(b²−4ac)) / (2a). It solves any quadratic equation ax²+bx+c=0 where a ≠ 0. The ± sign gives two roots: add √ for x₁ and subtract √ for x₂.

The discriminant Δ = b²−4ac: positive means two distinct real roots (parabola crosses x-axis twice), zero means one repeated root (parabola just touches x-axis), negative means complex roots (parabola does not touch x-axis).

The vertex is the turning point at (−b/2a, f(−b/2a)). When a > 0 the parabola opens upward and the vertex is the minimum. When a < 0 it opens downward and the vertex is the maximum.

Find the roots x₁ and x₂ using the quadratic formula, then write a(x − x₁)(x − x₂). For example, x²−5x+6=0 has roots x=2 and x=3, so factored form is (x−2)(x−3).

The axis of symmetry is the vertical line x = −b/(2a). The parabola is perfectly symmetric about this line, and the vertex lies on it. The two roots are equidistant from this line.