Equation Coefficients

x² - 5x + 6 = 0

Coefficient a (x² term):

Coefficient b (x term):

Coefficient c (constant):

Quadratic Formula:

x = (-b ± √(b² - 4ac)) / 2a

• a ≠ 0 (otherwise not quadratic)
• Discriminant = b² - 4ac
• Two real roots if discriminant > 0
• One real root if discriminant = 0
• Complex roots if discriminant < 0

Solution Results

Discriminant (Δ):

Root 1 (x₁):

Root 2 (x₂):

Vertex (h, k):

(2.5, -0.25)

Axis of Symmetry:

x = 2.5

Y-intercept:

Step-by-Step Solution:

1. Identify coefficients:

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

2. Calculate discriminant:

Δ = (-5)² - 4(1)(6) = 25 - 24 = 1

3. Apply quadratic formula:

x = (5 ± √1) / 2 = (5 ± 1) / 2

4. Calculate roots:

x₁ = 3, x₂ = 2

Parabola Graph

Graph Properties:

Opens: Upward

Vertex: (2.5, -0.25)

X-intercepts: x = 2, x = 3

Y-intercept: y = 6

Solution Type:

Two distinct real roots (Δ > 0)

Understanding Quadratic Equations and Polynomial Solutions

What is a Quadratic Equation?

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a ≠ 0. These equations appear frequently in physics, engineering, economics, and many real-world applications involving parabolic relationships and optimization problems.

Quadratic Formula

The quadratic formula x = (-b ± √(b² - 4ac)) / 2a provides the exact solutions for any quadratic equation. The discriminant (b² - 4ac) determines the nature of the roots: positive for two real roots, zero for one repeated root, negative for complex roots.

Solution Methods

Quadratic Formula: Universal method for all cases
Factoring: When equation factors nicely
Completing the Square: Converting to vertex form
Graphing: Visual solution method
Square Root Property: For perfect squares
Substitution: For complex equations

Discriminant Analysis

The discriminant Δ = b² - 4ac reveals solution characteristics: Δ > 0 yields two distinct real roots, Δ = 0 gives one repeated real root, and Δ < 0 produces two complex conjugate roots.

Quadratic Equation Applications

Physics and Engineering

Projectile motion, free fall calculations, optimization problems, electrical circuit analysis, structural engineering, and mechanical systems often involve quadratic relationships requiring equation solving.

Business and Economics

Profit maximization, cost minimization, revenue optimization, break-even analysis, supply and demand modeling, and financial planning frequently use quadratic equations for decision-making.

Education and Research

Students, teachers, and researchers use quadratic solvers for homework, problem verification, concept understanding, mathematical modeling, and academic research in algebra and calculus courses.

Mathematical Theory and Problem-Solving Strategies

Vertex Form and Transformations

The vertex form y = a(x - h)² + k reveals the parabola's vertex (h, k) and transformations. Converting between standard form ax² + bx + c and vertex form helps understand the function's geometric properties and optimization points.

Factoring Techniques

Factoring methods include finding common factors, grouping, difference of squares, perfect square trinomials, and trial-and-error. When factoring is possible, it provides the most elegant solution method.

Complex Number Solutions

When the discriminant is negative, solutions are complex numbers in the form a + bi. These complex roots are conjugate pairs and have important applications in electrical engineering, signal processing, and advanced mathematics.

Real-World Modeling

Quadratic equations model parabolic trajectories, optimization problems, area and volume relationships, profit functions, and many natural phenomena where variables have quadratic relationships.