# Quadratics

Quadratics are mathematical expressions, often represented as axÂ² + bx + c = 0, where ‘a’ is not equal to zero. Quadratic equations describe a specific type of curve in a graphical representation called a parabola. This mathematical construct is essential in physics to depict motion trajectories, in business to calculate profit and loss, and in machine learning algorithms for predictive modeling.

## Solving Quadratics

A quadratic expression is any expression with an x squared term, an x term and a constant. For example, is a quadratic expression. Note that it doesn’t have to be an , it could by or any other letter as long as it is the same throughout. Furthermore, there are three ways in which you can solve quadratics – each method requires setting the quadratic to 0 first. See Example 1.

### Factorising

Firstly, the simplest method, provided that it is possible, is factorising.

**Example:*** *

.

### Quadratic Formula

If factorising doesn’t work but a quadratic does have roots, the quadratic formula will find them instead. Recall that the discriminant will tell you how many roots a quadratic has. See Discriminants page.

The quadratic formula says that if then the roots are given by:

**Example:** In the following quadratic, , and :

given exactly, i.e. not as a rounded decimal. Rounded to two decimal places using a calculator, the solutions are and . See Example 2.

### Completing the Square

Alternatively, another infallible method for finding roots (if a quadratic is solvable) is to complete the square. See Completing the Square page and Example 3.

**Example:*** *

### Sketching Quadratics

It is worth noting that completing the square is also useful for sketching a quadratic. The reason for this is that, by writing the quadratic in completed square form, we can see the transformations applied to the graph of (the shape of a quadratic is a known as a parabola). For example, to get we shift the graph of by 3 ( transformation) and then up by 1 ( transformation).

**1**. Firstly, find the **roots **using one of the above methods, roots occur when .

**2.** Then, find the **-intercept**, this occurs when .

**3.** Finally, find the coordinates of the **vertex **by completing the square and applying transformations to .

See Completing the Square for more details and check out Example 4.

### DESMOS

DESMOS is a fantastic sketching tool. Click here to try it out. Firstly click the start graphing button and type y=x^2+4x-5 in the bar where the cursor starts. Then try adding more graphs and experimenting with the options. Finally, try exporting your graphs.

## Examples

## Videos

Sketching and finding the discriminant of a quadratic in a completing the square style question.

A real life quadratics question rooted in the context of profit versus selling price.