Quadratics

Quadratics are mathematical expressions, often represented as ax² + 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, $3x^2-4x+7$ is a quadratic expression. Note that it doesn’t have to be an $x$ , it could by $y$ 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:

$\begin{array}{lll}2x^2-5x-3=0\hspace{5pt}&\Rightarrow&\hspace{5pt}(2x+1)(x-3)=0\hspace{5pt}\\&\Rightarrow&\hspace{5pt} x=-\frac{1}{2}, x=3\end{array}$ .

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 $ax^2+bx+c=0$ then the roots are given by:

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Example: In the following quadratic, $a=1$ , $b=3$ and $c=-3$ :

$\begin{array}{lll}x^2+3x-3=0\hspace{5pt}&\Rightarrow&\hspace{5pt}x=\frac{-3\pm\sqrt{3^2-4\times 1\times-3}}{2}\hspace{5pt}\\&\Rightarrow&\hspace{5pt}x=\frac{-3 +\sqrt{21}}{2}, \frac{-3-\sqrt{21}}{2}\end{array}$

given exactly, i.e. not as a rounded decimal. Rounded to two decimal places using a calculator, the solutions are $x=0.79$ and $x=-3.79$ . 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:

$\begin{array}{lll}x^2+6x+5=0\hspace{3pt}\hspace{5pt}&\Rightarrow&\hspace{5pt}(x+3)^2-4=0\hspace{5pt}\\&\Rightarrow&\hspace{5pt} x+3=\pm 2\hspace{3pt}\hspace{5pt}\\&\Rightarrow&\hspace{5pt} x=-1, x=-5\end{array}$

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 $x^2$ (the shape of a quadratic is a known as a parabola). For example, to get $y=(x+3)^2+1$ we shift the graph of $x^2$ by 3 ( $x$ transformation) and then up by 1 ( $y$ transformation).

1. Firstly, find the roots using one of the above methods, roots occur when $y=0$ .

2. Then, find the $y$ -intercept, this occurs when $x=0$ .

3. Finally, find the coordinates of the vertex by completing the square and applying transformations to $y=x^2$ .

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.