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Quadratic Equation Solver

Solve ax2+bx+c=0ax^2 + bx + c = 0, see the parabola graph, roots, vertex, and step-by-step solution. Updated in real-time.

x2=0x^2 = 0

Results

Roots(One Repeated Root)
x=0x = 0
Vertex
(0, 0)\left(0,\, 0\right)
Axis of Symmetry
x=0x = 0
Y-Intercept
(0, 0)\left(0,\, 0\right)
Discriminant
Δ=0\Delta = 0

Open this tool with your own values

This tool reads its inputs from the page address, so a link can open it with the values from a problem already entered. Include every value the problem needs, because one left out keeps the tool's default. Use plain numbers in the stated units.

a
coefficient of x² in ax² + bx + c = 0
b
coefficient of x
c
constant term

Example link

Reference Guide

The Three Forms

Standard Form y=ax2+bx+cy = ax^2 + bx + c

Easiest to identify a, b, c for the quadratic formula.

Vertex Form y=a(x−h)2+ky = a(x - h)^2 + k

Directly shows the vertex (h, k) and direction of opening.

Factored Form y=a(x−r1)(x−r2)y = a(x - r_1)(x - r_2)

Directly shows the roots. Only possible when roots are real.

The Quadratic Formula

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Use this when the equation can't be easily factored.

Steps

  1. Identify a, b, and c from standard form
  2. Calculate the discriminant Δ=b2−4ac\Delta = b^2 - 4ac
  3. Substitute into the formula and simplify

Understanding the Discriminant

Δ=b2−4ac\Delta = b^2 - 4ac
Δ>0\Delta > 0 Two distinct real roots. Parabola crosses x-axis twice.
Δ=0\Delta = 0 One repeated root. Parabola touches x-axis at vertex.
Δ<0\Delta < 0 Two complex roots. Parabola doesn't cross x-axis.

Vertex & Axis of Symmetry

Vertex formula h=−b2a,k=f(h)h = \frac{-b}{2a}, \quad k = f(h)
Axis of symmetry The vertical line x=hx = h

a>0a > 0 Parabola opens upward, vertex is the minimum

a<0a < 0 Parabola opens downward, vertex is the maximum

TopicsAlgebraGraphsFunctionsProcessMathematics

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