Quadratic Formula Calculator

Solve quadratic equations of the form ax² + bx + c = 0 using the quadratic formula. Find roots, vertex, discriminant, and graph properties with step-by-step solutions.

Quadratic Equation: ax² + bx + c = 0

Coefficient a (x² term)

Coefficient b (x term)

Coefficient c (constant term)

Your Equation:

x² - 5x + 6 = 0

Quick Examples

Discriminant Analysis

Discriminant (Δ):1

Two Distinct Real Roots

The parabola intersects the x-axis at two points

Formula: Δ = b² - 4ac

Δ > 0: Two distinct real roots
Δ = 0: One repeated real root
Δ < 0: Two complex conjugate roots

Solutions

Roots:

x₁ = 3.0000

x₂ = 2.0000

Sum of roots:5.0000

Product of roots:6.0000

Parabola Properties

Vertex:(2.5000, -0.2500)

Axis of symmetry:x = 2.5000

Opens:Upward

Y-intercept:(0, 6)

Step-by-Step Solution

Step 1: Identify coefficients

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

Step 2: Calculate discriminant

Δ = b² - 4ac = -5² - 4(1)(6) = 1

Step 3: Apply quadratic formula

x = (-b ± √Δ) / (2a) = (--5 ± √1) / (2 × 1)

Step 4: Calculate roots

x₁ = 3.0000

x₂ = 2.0000

How to Use This Calculator

This quadratic formula calculator solves equations of the form ax² + bx + c = 0 and provides comprehensive analysis.

Enter the coefficients a, b, and c for your quadratic equation
The calculator will automatically compute the discriminant and determine the nature of roots
View the solutions, vertex, axis of symmetry, and other parabola properties
Follow the step-by-step solution to understand the calculation process
Use quick examples to see how different types of quadratic equations are solved

Understanding the Quadratic Formula

The Formula

The quadratic formula is: x = (-b ± √(b² - 4ac)) / (2a)

Key Components

Discriminant (Δ): b² - 4ac determines the nature of roots
Vertex: The turning point of the parabola at (-b/2a, f(-b/2a))
Axis of Symmetry: The vertical line x = -b/2a
Roots: The x-intercepts where the parabola crosses the x-axis

Types of Solutions

Two Real Roots: When Δ > 0, the parabola crosses the x-axis twice
One Real Root: When Δ = 0, the parabola touches the x-axis once (vertex on x-axis)
Complex Roots: When Δ < 0, the parabola doesn't cross the x-axis

Vieta's Formulas

Sum of roots: x₁ + x₂ = -b/a
Product of roots: x₁ × x₂ = c/a

Example Calculations

Example 1: Two Real Roots

Solve: x² - 5x + 6 = 0

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

Δ = (-5)² - 4(1)(6) = 25 - 24 = 1

x = (5 ± √1) / 2 = (5 ± 1) / 2

x₁ = 3, x₂ = 2

Example 2: One Repeated Root

Solve: x² - 4x + 4 = 0

Solution: a = 1, b = -4, c = 4

Δ = (-4)² - 4(1)(4) = 16 - 16 = 0

x = 4 / 2 = 2 (repeated root)

Example 3: Complex Roots

Solve: x² + x + 1 = 0

Solution: a = 1, b = 1, c = 1

Δ = 1² - 4(1)(1) = 1 - 4 = -3

x = (-1 ± √(-3)) / 2 = (-1 ± i√3) / 2

Frequently Asked Questions

What if the coefficient 'a' is zero?

If a = 0, the equation becomes linear (bx + c = 0) rather than quadratic. The quadratic formula doesn't apply, and the solution is simply x = -c/b (provided b ≠ 0).

How do I interpret complex roots?

Complex roots occur when the discriminant is negative. They come in conjugate pairs (a + bi and a - bi) and indicate that the parabola doesn't intersect the x-axis. These roots are still mathematically valid solutions.

Can I use this for incomplete quadratic equations?

Yes! If b = 0 or c = 0, simply enter 0 for that coefficient. For example, x² - 4 = 0 would have a = 1, b = 0, c = -4.