## Introduction

Quadratic equations are a fundamental part of algebra and are essential for understanding various aspects of mathematics and science. Using tools like a Quadratic Calculator or understanding the Quadratic Formula can simplify the process of finding roots.

## Definition

A quadratic equation is any equation that can be written in the standard form `ax^2 + bx + c = 0`

, where `a`

, `b`

, and `c`

are constants, and `a`

is not equal to 0.

## Roots of a Quadratic Equation

The solutions to a quadratic equation are known as the roots of the equation. These are the values of `x`

that make the equation true.

### Types of Roots

**Real and Equal Roots**: Occurs when the discriminant `b^2 - 4ac`

is zero.
**Real and Distinct Roots**: Occurs when the discriminant is positive.
**Complex Roots**: Occurs when the discriminant is negative.

## Methods of Solving Quadratic Equations

### Quadratic Formula

The most universal method for solving a quadratic equation is the Quadratic Formula. It is given by:

`x = (-b +/- sqrt(b^2 - 4ac)) / (2a)`

### Factorising

The equation can be converted into a product of two binomial expressions, which are then individually set to zero to find the values of `x`

.

### Completing the Square

This method involves turning a quadratic equation into a perfect square trinomial, thus simplifying the solving process.

### Quadratic Calculator

For those who need a quick and efficient way to find roots, a Quadratic Calculator can be extremely helpful.

## Applications

Quadratic equations are used in various real-world scenarios:

**Physics**: In projectile motion, optimising paths.
**Engineering**: In structural design, electrical circuits.
**Finance**: For calculating interest and investment growth.

## Graphical Representation

A quadratic equation is graphically represented by a parabola. The orientation (upwards or downwards) depends on the coefficient `a`

.

### Vertex

The vertex is the highest or lowest point of the parabola, given by `(-b / (2a), f(-b / (2a)))`

.

### Axis of Symmetry

The axis of symmetry is a vertical line passing through the vertex, and its equation is `x = -b / 2a`

.

## Conclusion

Quadratic equations are a cornerstone in the field of algebra. Understanding their properties and how to solve them using methods like the Quadratic Formula or a Quadratic Calculator sets the foundation for more advanced mathematical and scientific study.