Formula
x = (-b ± √(b² - 4ac)) ÷ 2a. Discriminant D = b² - 4ac. Vertex x-coordinate = -b ÷ 2a.
Quadratic Formula Calculator
Solve ax^2 + bx + c = 0 with the quadratic formula, discriminant, roots, vertex check and printable working.
What-if check
Discriminant-first proof
The discriminant decides the root type before the quadratic formula is evaluated, so the printout preserves that checkpoint.
| Check | Value | Note |
|---|---|---|
| a, b, c | 1, -3, 2 | quadratic coefficients |
| discriminant | 1 | two real roots |
| vertex x | 1.5 | -b divided by 2a |
| current output | x = 2 or 1 | formula result |
Visual proof
Result path
Discriminant 1; roots = (-b ± √discriminant) ÷ 2a. Vertex (1.5, -0.25).
Worked example
For x² - 3x + 2 = 0, D = (-3)² - 4(1)(2) = 1. Roots are (3 ± 1) ÷ 2, so x = 2 or x = 1.
Professional note
Master’s Tip: check the discriminant first. Positive means two real roots, zero means one repeated real root, and negative means a complex pair.
Regional and unit assumptions
Standard or basis: ordinary real/complex algebra for second-degree polynomial equations; no curriculum-specific rounding rule is claimed.
Assumptions and limitations
- Inputs are numeric coefficients in ax² + bx + c = 0.
- If a is zero, the page falls back to the linear equation bx + c = 0.
- Negative discriminants are shown as complex conjugate roots.
- Displayed roots are rounded; formula steps keep the coefficient trail visible.
Methodology & Accuracy
How this calculator is checked
CalculationTime pages are built around visible arithmetic: the formula, assumptions, worked example and practical limitations are shown so the result can be checked rather than simply trusted.
Formula used
x = (-b ± √(b² - 4ac)) ÷ 2a. Discriminant D = b² - 4ac. Vertex x-coordinate = -b ÷ 2a.
Standard or basis
Standard or basis: ordinary real/complex algebra for second-degree polynomial equations; no curriculum-specific rounding rule is claimed.
Where a calculator follows a named legal, trade or industry standard, that standard is cited visibly. Otherwise the page uses transparent general arithmetic and states its limits.Master's Tip
Master’s Tip: check the discriminant first. Positive means two real roots, zero means one repeated real root, and negative means a complex pair.
Related calculators
Questions
What is the quadratic formula?
The quadratic formula is x = (-b ± √(b² - 4ac)) ÷ 2a for equations written as ax² + bx + c = 0.
What does the discriminant tell me?
The discriminant b² - 4ac tells whether the roots are two real roots, one repeated real root, or a complex pair.
What if a equals zero?
Then the equation is not quadratic. This page solves the remaining linear equation when b is not zero.
Calculation note
Quadratic equations connect algebra, geometry and graphing. The discriminant and vertex make the answer more useful than a bare pair of roots.
The discriminant is the first checkpoint
Before calculating roots, the discriminant explains the shape of the answer. It prevents confusion when a calculator returns complex values.
The vertex explains the graph
The same coefficients also define the parabola vertex. Showing it beside the roots helps connect equation solving to graph behavior.