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.

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:

.

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:

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

1.
2.

Solution:

1. Write as and then factorise to . The two factors and could both be zero to give a product of 0 and so the solutions are or .
2. First write  as and then divide both sides by 2 to give . Then factorise to and so the solutions are and .

Solution:

Since the question says find the solutions to two decimal places, it suggests that we should use the quadratic formula. Selecting , and , the solutions are given by

Solve the equation by completing the square. Give exact answers, simplified where appropriate.

Solution:

Taking out a factor of 2 gives the equation:

.

Hence, and .

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.