Quadratic Formula Calculator

Solve any quadratic equation using the quadratic formula. Enter coefficients a, b, c to find roots, discriminant, vertex, axis of symmetry, and parabola graph. Shows step-by-step solutions, Vieta's formulas, standard/vertex/factored forms. Handles real and complex roots with exact and decimal answers.

Coefficients
x² - 5x + 6 = 0
Two Distinct Real Roots
x₁ =3
x₂ =2
≈ 3, 2
Δ = 1
Vertex (2.5, -0.25)
Opens Up

Key Properties

Discriminant, Vieta's formulas & parabola

Discriminant (\u0394)
1
Sum of Roots
5
Product of Roots
6
Vertex
(2.5, -0.25)
Axis of Symmetry
x = 2.5
Y-Intercept
(0, 6)
Direction
Upward (a > 0)
Minimum Value
-0.25

Equation Forms

Standard, vertex & factored representations

Standard Form
x² - 5x + 6 = 0
Vertex Form
(x - 2.5)² - 0.25 = 0
Factored Form
(x - 3)(x - 2) = 0

Parabola Graph

Visualize the parabola, roots, and vertex on the coordinate plane

Parabola
Vertex
Roots
Y-Intercept

Step-by-Step Solution

Detailed walkthrough of the quadratic formula applied to your equation

1

Identify coefficients

ax² + bx + c = 0

a = 1, b = -5, c = 6

x² - 5x + 6 = 0

2

Calculate the discriminant

Δ = b² − 4ac

Δ = (-5)² − 4(1)(6)

Δ = 25 − 24 = 1

3

Discriminant is positive

Δ > 0 → Two distinct real roots

Δ = 1 > 0

The equation has two distinct real roots.

4

Apply the quadratic formula

x = (−b ± √Δ) / 2a

x = (−(-5) ± √(1)) / (2 × 1)

x = (5 ± √(1)) / 2

5

Compute the roots

x₁ = (−b + √Δ) / 2a, x₂ = (−b − √Δ) / 2a

x₁ = (5 + 1) / 2, x₂ = (5 − 1) / 2

x₁ = 3, x₂ = 2

6

Verify with Vieta's formulas

x₁ + x₂ = −b/a, x₁ × x₂ = c/a

Sum = −(-5)/1 = 5, Product = 6/1 = 6

Sum of roots = 5, Product of roots = 6

What Is the Quadratic Formula?

The universal formula for solving quadratic equations

A quadratic equation is any equation of the form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. The quadratic formula provides a direct way to find the solutions (roots) of any quadratic equation, regardless of whether it can be factored.

The Quadratic Formula

x = (−b ± √(b² − 4ac)) / 2a

The expression under the square root, b² − 4ac, is called the discriminant (Δ). It determines the nature of the roots: positive means two distinct real roots, zero means one repeated root, and negative means two complex conjugate roots.

Understanding the Discriminant

How Δ = b² − 4ac determines root type

Δ > 0 — Two distinct real roots

The parabola crosses the x-axis at two points. Example: x² − 5x + 6 = 0 has Δ = 1, giving x = 2 and x = 3.

Δ = 0 — One repeated (double) root

The parabola touches the x-axis at exactly one point (the vertex). Example: x² − 6x + 9 = 0 has Δ = 0, giving x = 3 (repeated).

Δ < 0 — Two complex conjugate roots

The parabola does not cross the x-axis. Example: x² + 2x + 5 = 0 has Δ = −16, giving x = −1 ± 2i.

Vieta's Formulas & Equation Forms

Useful relationships between coefficients and roots

Vieta's formulas relate the coefficients of a quadratic to the sum and product of its roots without solving the equation:

x₁ + x₂ = −b/a

x₁ × x₂ = c/a

A quadratic equation can be expressed in three equivalent forms:

  • Standard form: ax² + bx + c = 0 — directly shows coefficients.
  • Vertex form: a(x − h)² + k = 0 — shows the vertex (h, k) of the parabola.
  • Factored form: a(x − r₁)(x − r₂) = 0 — shows the roots directly (only when roots are real).

Common Mistakes to Avoid

Pitfalls students frequently encounter

  • Forgetting to set the equation to zero — the formula only works when one side equals 0. Rearrange first.
  • Sign errors with −b — when b is already negative, −b becomes positive. Use parentheses carefully.
  • Dividing by 2 instead of 2a — the entire denominator is 2a, not just 2. If a = 3, divide by 6.
  • Forgetting both ± solutions — the ± means you must compute both the + and − cases, giving two roots.
  • Not simplifying radicals — always reduce √(b² − 4ac) to simplest form, e.g. √72 = 6√2.

Real-World Applications

Where quadratic equations appear in practice

  • Projectile motion: The height of a thrown ball follows h(t) = −½gt² + v₀t + h₀. Solving for h = 0 gives the time of impact.
  • Area optimization: Finding the dimensions of a rectangle with maximum area for a given perimeter involves solving a quadratic.
  • Profit maximization: Revenue and cost models in economics often produce quadratic profit functions; the vertex gives maximum profit.
  • Engineering: Parabolic shapes appear in satellite dishes, suspension bridge cables, and lens design, all modeled by quadratic equations.

Frequently Asked Questions

Common questions about quadratic equations and the quadratic formula

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