Matemáticas y Álgebra

Free Quadratic Equation & Parabola Vertex Solver

Quadratic equations are fundamental to high school algebra, physics projectile trajectories, and geometric parabola analysis. Solving quadratic equations by hand requires applying the quadratic formula carefully to avoid sign errors. Our free quadratic equation solver computes solutions instantly. Find real and complex roots for ax² + bx + c = 0, evaluate discriminant Δ = b² - 4ac, and utilize our vertex of parabola calculator to locate quadratic peak and trough coordinates.

Quadratic Equation & Parabola Vertex Solver

Solve ax² + bx + c = 0 quadratic equations, calculate real/complex roots, discriminant Δ & parabola vertex

Enter Quadratic Coefficients:

x² +x += 0

Roots & Vertex Results

Roots Solution:

x₁ = 3 | x₂ = 2

Two Distinct Real Roots (Δ > 0)

Discriminant Δ:1

Parabola Vertex (h, k):(2.50, -0.25)

Step-by-Step Derivation:

Coefficient values: a = 1, b = -5, c = 6
Discriminant Δ = b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1
x₁ = (-b + √Δ) / 2a = (5 + 1) / 2 = 3
x₂ = (-b - √Δ) / 2a = (5 - 1) / 2 = 2
Parabola Vertex (h, k) = (2.5, -0.25)

Mathematical Formulas & Calculation Methods

Quadratic Formula & Discriminant

Solves roots for ax² + bx + c = 0 with discriminant Δ = b² - 4ac.

x = (-b ± √(b² - 4ac)) / 2a | Vertex (h, k): h = -b / 2a, k = c - b²/4a

Worked Step-by-Step Examples

Example 1: Solving x² - 5x + 6 = 0

"Find roots x1, x2, discriminant Δ, and parabola vertex."

Steps:

a = 1, b = -5, c = 6.
Discriminant Δ = (-5)² - 4(1)(6) = 25 - 24 = 1 (Δ > 0, two real roots).
x1 = (5 + √1) / 2 = 6 / 2 = 3.
x2 = (5 - √1) / 2 = 4 / 2 = 2.
Vertex h = -(-5) / 2(1) = 2.5; k = (2.5)² - 5(2.5) + 6 = -0.25 → (2.5, -0.25).

Result: x₁ = 3, x₂ = 2 (Vertex: (2.5, -0.25))

What is Solucionador de Quadratic Equation & Parabola Vertex?

A discriminant calculator b2 4ac determines whether a quadratic equation has two distinct real roots (Δ > 0), one repeated real root (Δ = 0), or two complex imaginary roots (Δ < 0).

The vertex of parabola calculator identifies the minimum or maximum peak point of a quadratic curve.

How to Use Solucionador de Quadratic Equation & Parabola Vertex

1Enter coefficients a, b, and c into the quadratic equation grid.
2Review calculated roots x1 & x2, discriminant Δ, and parabola vertex (h, k).

Key Features

Full support for real and complex imaginary roots.
Calculates discriminant Δ = b² - 4ac step-by-step.
Calculates parabola vertex (h, k) coordinates.

Benefits of Using Solucionador de Quadratic Equation & Parabola Vertex

Ideal for verifying algebra and calculus homework solutions.

Saves time in physics projectile motion problem solving.

Practical Use Cases

Algebra Students

Check homework steps for quadratic factoring and formula applications.

Frequently Asked Questions

What does a negative discriminant mean?

A negative discriminant (Δ < 0) means the quadratic equation has no real x-intercepts and instead yields two complex conjugate imaginary roots containing 'i'.

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