Quadratics - methods for solving and sketching - StudyWell

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.

1. Factorising

Firstly, the simplest method provided that it can be done, is factorising.

Example:

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

2. 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:

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

given exactly, i.e. not as a rounded decimal.

3. Completing the Square

Alternatively, another infallible method for finding roots if a quadratic can be solved is to complete the square. See Completing the Square page.

Example

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

Sketching Quadratics

It it 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 $x^2$ . For example, $y=(x+3)^2+1$ is the graph of $x^2$ shifted to the left by 3 (x transformation) and then up by 1 (y transformation).

quadratics

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.

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.

Quadratics Notes

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Quadratics Practice

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Click here to find Questions by Topic – scroll down to all past QUADRATIC questions.

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