Quadratic Equation Solver

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

x^2 = 0

a (leading coefficient)

b (linear coefficient)

c (constant term)

Roots(One Repeated Root)

x = 0

Vertex

\left(0,\, 0\right)

Axis of Symmetry

x = 0

Y-Intercept

\left(0,\, 0\right)

Discriminant

\Delta = 0

Reference Guide

Standard Form

y = ax^2 + bx + c

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

Vertex Form

y = a(x - h)^2 + k

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

Factored Form

y = a(x - r_1)(x - r_2)

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

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

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

Steps

\Delta = b^2 - 4ac

$\Delta > 0$ Two distinct real roots. Parabola crosses x-axis twice.

$\Delta = 0$ One repeated root. Parabola touches x-axis at vertex.

$\Delta < 0$ Two complex roots. Parabola doesn't cross x-axis.

Vertex formula

h = \frac{-b}{2a}, \quad k = f(h)

Axis of symmetry The vertical line

x = h

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

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