Certified calculator · Version 1

Quadratic Formula Calculator

Solve ax² + bx + c = 0 with the quadratic formula, including real or complex roots and deterministic intermediate steps.

Coefficient aCoefficient bConstant c

Roots3, 2

Discriminant1

Vertex x2.5

Vertex y-0.25

Trend

Included with Free

1.50.75

2.5-0.25

3.50.75

4.53.75

5.58.75

6.515.75

Visual scale

Accessible data view

-1.5

-0.5

0.5

1.5

2.5

3.5

4.5

5.5

6.5

CalcuMint Pro

Advanced Analysis

Compare plans

Advanced chartsExplore the result across time and assumptions.

Scenario comparisonCompare multiple assumptions side by side.

Sensitivity analysisSee which assumptions change the outcome most.

Goal SolverSolve backwards from the outcome you want.

Result methodology

The certified engine computes the discriminant, roots and vertex. The plotted sample points are evaluated from the same validated coefficients.

Solution steps

D = b² - 4ac = 1
Use stable q = -½(b + sign(b)√D)
x₁ = 3, x₂ = 2

How this calculator works

Compute D = b² - 4ac, then evaluate x = (-b ± √D)/(2a). Negative discriminants are represented as complex conjugate roots.

Formula

x = (-b ± √(b² - 4ac)) / (2a)

Uses the discriminant to determine and calculate the two roots of ax² + bx + c = 0.

Worked example

x² - 5x + 6 = 0 is a verified test case used by the calculation engine.

Assumptions and limitations

Coefficient a must be non-zero.
Inputs are finite real numbers.

Methodology & sources

This calculator uses deterministic, versioned calculation logic. The formula and verified examples above are part of the calculation definition used by CalcuMint.

CalcuMint deterministic algebra engine — Standard quadratic formula and vertex identities.

Frequently asked questions

Does it support complex roots?

Yes. A negative discriminant is returned as a pair of complex conjugate roots.

Does it calculate the vertex?

Yes. The deterministic output includes the vertex coordinates.