Calculat.dev LogoCalculat.dev - All Calculators, One Place.devAll Calculators, One Place.
Math CalculatorsPopular Tool Runs Locally

Quadratic Equation Calculator

Solve quadratic equations ax² + bx + c = 0 using the quadratic formula. Computes discriminant Δ, real and complex conjugate roots, vertex, and axis of symmetry.

Roots & Discriminant
x1 = 3x2 = 2Δ = 1 (two-real)
Parabola Vertex: (2.5, -0.25) | Axis of Symmetry: x = 2.5
Deduction Steps:
Standard form: ax² + bx + c = 0 → 1x² + (-5)x + (6) = 0
Discriminant: Δ = b² - 4ac = (-5)² - 4(1)(6) = 1
Vertex: (h, k) = (-b / 2a, f(h)) = (2.5, -0.25)
Δ > 0: Two distinct real roots.
x₁ = (-(-5) + √1) / (2 × 1) = 3
x₂ = (-(-5) - √1) / (2 × 1) = 2
🎓

Did this calculation save you time?

Student Project

Hi! I'm a student developer building Calculat in my spare time. I was tired of searching for basic math tools and having to click through 10 spammy popups, loan ads, and cookie trackers.

I keep this website 100% free, private, and ad-free. If this helped you with your homework, project, or finances today, bookmarking this page or telling a friend helps me keep building more free tools!

Press ⌘ + D to Bookmark
Suggest What I Build Next

What Is a Quadratic Equation Calculator?

A quadratic equation is a second-order polynomial equation of the standard form ax² + bx + c = 0, where a ≠ 0. The solutions are called roots or zeros, geometrically representing the points where the parabola crosses the x-axis.

How to Use This Calculator

  1. Enter coefficients a (for x²), b (for x), and constant c.
  2. View the discriminant Δ = b² - 4ac and classification of roots.
  3. Review the real roots or complex conjugate pairs, parabola vertex, and step-by-step algebra.

Quadratic Formula & Discriminant

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

If Δ > 0, there are two distinct real roots. If Δ = 0, there is one repeated real root. If Δ < 0, there are two complex conjugate roots.

Worked Example

Scenario: Solve 2x² - 4x - 6 = 0

Identify coefficients: a = 2, b = -4, c = -6

Compute discriminant: Δ = (-4)² - 4(2)(-6) = 16 + 48 = 64

Apply quadratic formula: x = (4 ± √64) / (2 × 2) = (4 ± 8) / 4

x₁ = (4 + 8)/4 = 3, x₂ = (4 - 8)/4 = -1

Result: x₁ = 3, x₂ = -1

Tips & Key Notes

  • The vertex of the parabola always occurs at x = -b / (2a).
  • When b = 0, the equation reduces to ax² + c = 0, which can be solved directly by taking square roots: x = ±√(-c/a).
  • If the discriminant is a perfect square, the quadratic can be factored over rational numbers.

Frequently Asked Questions

What does a negative discriminant mean?

When Δ < 0, the square root involves a negative number, meaning the parabola does not touch the real x-axis and the roots are complex conjugate numbers (p ± qi).

Why must coefficient a not equal 0?

If a = 0, the x² term disappears, turning the quadratic equation into a first-degree linear equation (bx + c = 0).

How do you find the vertex of a parabola from coefficients?

The x-coordinate of the vertex is h = -b / (2a), and the y-coordinate is k = c - b² / (4a).

What is the axis of symmetry?

The axis of symmetry is the vertical line x = -b / (2a) that divides the parabola into two mirrored halves.

Related Calculators

Explore similar tools
PopularMath

Linear Equation Calculator

Solve linear algebraic equations of the form ax + b = c with full step-by-step mathematical deduction and solution classification.

Calculate
PopularMath

System of Linear Equations Calculator

Solve simultaneous 2x2 and 3x3 systems of linear equations using Cramer’s Rule and matrix determinants with step-by-step work.

Calculate
PopularMath

Square Root Calculator

Calculate the square root of positive and negative numbers (√x). Provides radical simplification, perfect square check, and imaginary roots.

Calculate