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Math & Numbers Runs in browser Updated 30 Mar 2026

Quadratic Equation Solver

Solve ax² + bx + c = 0 with real or complex roots, discriminant, vertex form, axis of symmetry, and a step-by-step parabola graph.

1x² + 0x + 0 = 0

x₁

-0

x₂

-0

Discriminant (b² − 4ac)

One repeated real root

0

Vertex

(-0, 0)

Axis of Symmetry

x = -0

Y-Intercept

(0, 0)

Parabola Graph

vertex

Step-by-Step Solution

1. Equation

1x² + 0x + 0 = 0

2. Discriminant

D = b² − 4ac = (0)² − 4(1)(0)

D = 0 − 0 = 0

3. Apply Quadratic Formula

x = (−b ± √D) / 2a

x = (-0 ± √0) / 2

√0 = 0

x₁ = (-0 + 0) / 2 = -0

x₂ = (-0 − 0) / 2 = -0

4. Vertex & Axis of Symmetry

h = −b / 2a = -0 / 2 = -0

k = c − b² / 4a = 0 − 0 / 4 = 0

Vertex: (-0, 0) — Axis: x = -0

Vertex form: 1(x − -0)² + 0

Quadratic Formula

x = (−b ± √(b² − 4ac)) / 2a

where a, b, and c are the coefficients of ax² + bx + c = 0.
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How to use Quadratic Equation Solver

  1. Enter coefficient a

    Type the value of a (the x² coefficient). Must be nonzero for a quadratic equation.

  2. Enter coefficient b

    Type the value of b (the x coefficient). Can be zero.

  3. Enter coefficient c

    Type the constant term c.

  4. Read the roots

    The roots x₁ and x₂ appear instantly with the discriminant. Complex roots show in a + bi form.

  5. Expand step-by-step solution

    Click 'Show steps' to see the full quadratic formula substitution and simplification.

  6. Read the parabola graph

    The SVG graph shows the parabola with roots and vertex labeled. Adjust coefficients to watch it update live.

Quadratic Equation Solver FAQ

What is the quadratic formula?
x = (−b ± √(b²−4ac)) / 2a. It gives both roots of ax²+bx+c=0. The ± means there are two solutions: one with addition and one with subtraction.

What is the discriminant?
The discriminant is b²−4ac. If positive: two distinct real roots. If zero: one repeated real root. If negative: two complex conjugate roots.

What if the discriminant is negative?
The equation has two complex roots in the form a ± bi where i = √(−1). The parabola does not cross the x-axis.

What is the vertex of a parabola?
The vertex is the highest or lowest point: h = −b/2a, k = f(h). Vertex form: a(x−h)²+k. The axis of symmetry is the vertical line x = h.

What is vertex form?
a(x−h)²+k, where (h,k) is the vertex. It shows the parabola's turning point directly and is useful for graphing.

What if a equals zero?
When a=0, the equation is linear (bx+c=0) with one root: x = −c/b. The solver handles this case automatically.

Can I solve equations with decimal coefficients?
Yes. Enter any real number including decimals for a, b, and c.

What are the roots of a quadratic?
The roots (also called zeros or x-intercepts) are the values of x where the parabola crosses the x-axis, i.e. where ax²+bx+c=0.

Can I use this for homework?
Yes. The step-by-step solution shows every stage of the quadratic formula substitution so you can verify your work.

Is any data sent to a server?
No. All calculations run entirely in your browser. Nothing is uploaded or stored.

Background

Solve any quadratic equation of the form ax² + bx + c = 0. Enter the coefficients a, b, and c to instantly get both roots, including complex roots when the discriminant is negative. See the discriminant value, the vertex coordinates, axis of symmetry, y-intercept, and vertex form alongside a step-by-step substitution into the quadratic formula. An SVG parabola graph plots the curve with roots and vertex labeled — a live visual that updates as you change coefficients. If a equals zero, the equation is treated as linear. Useful for algebra and pre-calculus students, physics problems involving projectile motion, engineering optimization, and anyone who needs roots of a second-degree polynomial. All processing runs in your browser with no software needed.

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