Quadratic Equation Calculator Australia 2026-27

Q: Can I factorise instead?

When the roots are rational, yes: x² − 3x + 2 = (x − 1)(x − 2). The formula always works, factorising only sometimes.

GlobalCalc

Inputs

a (coefficient of x²)

b (coefficient of x)

c (constant)

Results update as you type

Results

Roots

x = 2 or x = 1

Discriminant (b² − 4ac)—

Nature of roots—

Vertex—

Sum and product of roots—

Reviewed September 2026. Pure mathematics: the result does not depend on where you are. Terminology follows the Australian Curriculum (maths, brackets, decimal point).

All calculations run 100% in your browser. The calculator code does not submit your figures to GlobalCalc to obtain a result.

About quadratic equation

Method

How the quadratic equation calculator works

The quadratic formula gives the roots: x = (−b ± √(b² − 4ac)) ÷ 2a. The discriminant b² − 4ac decides the shape of the answer: positive → two real roots, zero → one repeated root, negative → two complex roots. The parabola's vertex is at x = −b ÷ 2a.

Formula: x = (−b ± √(b² − 4ac)) / 2a

Data

Worked examples

Inputs	Roots	Note
x² − 3x + 2 = 0	x = 2 or x = 1	roots 1 and 2
x² − 4x + 4 = 0	x = 2	repeated root 2
x² + x + 1 = 0	x = -0.5 ± 0.866025i	complex roots
2x² + 5x − 3 = 0	x = 0.5 or x = -3	roots 0.5 and −3

FAQ

Frequently asked questions

What is the discriminant?

b² − 4ac. Its sign tells you whether the roots are two real numbers, one repeated number or a complex pair.

What does a complex root mean?

The parabola never crosses the x-axis. The roots still exist as complex numbers a ± bi.

How do I complete the square?

Rewrite ax² + bx + c as a(x + b/2a)² + (c − b²/4a); the vertex is then obvious.

What is the vertex?

The turning point of the parabola, at x = −b/2a. It is a minimum when a > 0 and a maximum when a < 0.

Can I factorise instead?

When the roots are rational, yes: x² − 3x + 2 = (x − 1)(x − 2). The formula always works, factorising only sometimes.

Sources

Where these figures come from

NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
Wolfram MathWorld — definitions and formulas for every topic on this page
NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
Australian Curriculum (ACARA) — Mathematics — the terms and methods taught in Australian schools

Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.