x²

Math & Statistics

Quadratic Equation Calculator

Solve a quadratic equation from its coefficients, identify the discriminant and root type, inspect the vertex and axis of symmetry, review a value table, visualize the parabola, and export a professional PDF analysis.

Coefficient a Coefficient b Coefficient c

Root 1-

Root 2-

Discriminant and root type-

Vertex-

Axis of symmetry-

Function analysis

Parabola around the vertex

Quadratic functionSampled around the vertex and real roots

x	ax2	bx	c	f(x)

Decision view

Parabola, vertex, and real-root map

The entered quadratic is drawn around its vertex with real roots and the y-intercept identified when visible.

Equation-

Vertex-

Real x-intercepts-

How to use Quadratic Equation Calculator

Enter coefficients a, b, and c for ax² + bx + c = 0, ensuring a is not zero.
Use the discriminant to identify whether the equation has two real roots, one repeated root, or two complex roots.
Read the vertex and axis of symmetry, then inspect the parabola and value table as an independent check.
Substitute each reported real root back into the equation when exact verification is important.

Calculator guide

Understanding Quadratic Equation Calculator

A quadratic equation describes a parabola and can have two real roots, one repeated real root, or a complex-conjugate pair. The discriminant connects the algebraic answer with the graph's relationship to the x-axis.

Discriminant The value b squared minus 4ac determines the number and type of roots.

Roots Values of x where the quadratic expression equals zero.

Vertex The minimum or maximum point of the parabola.

Axis of symmetry The vertical line through the vertex at x = -b/(2a).

Calculation method

How the calculation works

For ax^2 + bx + c = 0, x = (-b +/- sqrt(b^2 - 4ac)) / (2a). Confirm that a is nonzero, calculate the discriminant, apply the quadratic formula, and calculate the vertex from -b/(2a). The value table samples points around the vertex for the graph.

Function geometry

How coefficients shape the parabola

The coefficients control opening direction, width, horizontal position, vertical position, and the existence of real x-intercepts.

Coefficient a Its sign controls whether the parabola opens upward or downward; magnitude affects width.

Coefficient b Together with a, it determines the axis of symmetry x = −b/(2a).

Coefficient c The y-intercept is (0,c).

Discriminant b² − 4ac determines the count and type of algebraic roots.

Verification

Three ways to check a solution

A reliable result should agree algebraically, numerically, and geometrically.

Substitution Insert a reported root into ax² + bx + c; the result should be zero within rounding.

Vertex symmetry When two real roots exist, their midpoint equals the axis of symmetry.

Graph intercepts Real roots appear where the parabola crosses or touches the x-axis.

Rounded displayed roots can leave a small substitution residual even when the full-precision calculation is correct.

Worked situations

Practical examples

Solve an equation with two distinct real roots.
Inspect a repeated root when the discriminant is zero.
Review complex roots and the real-valued parabola when the discriminant is negative.

Better inputs

Useful tips

Keep coefficient signs exactly as written.
Use the discriminant to check the reported root type.
Substitute real roots back into the original equation when precision matters.

Before relying on the result

Limitations and common mistakes

The coefficient a cannot be zero because the equation would be linear.
Displayed decimal roots are rounded approximations.
The graph uses an automatically selected window and may not show distant features.

Reference

Key terms

Coefficient: A numerical multiplier attached to a variable term.
Discriminant: b squared minus 4ac.
Root: An x-value that makes the expression zero.
Vertex: The turning point of the parabola.

Important note

Calculated directly from the entered values using the displayed formula and rounding settings.

Frequently asked questions

What does a negative discriminant mean?

The equation has two complex-conjugate roots and no real x-axis intersections.

Why must a be nonzero?

If a is zero, the squared term disappears and the equation is not quadratic.

When is there one repeated root?

A repeated real root occurs when the discriminant is exactly zero.

Does the graph prove the roots?

The graph is a visual aid; the quadratic formula supplies the numerical roots.