# How do you solve x^2+6=0 by graphing?

May 31, 2017

See below.

#### Explanation:

You make the equation into one or two functions.

Starting note: I will use a standard coordination system, with y- and x-axis, this means $f \left(x\right) = y$ and $g \left(x\right) = y$

First look at the left side. Create the function $f \left(x\right) = {x}^{2} + 6$ , this means we no longer limit the expression to one particular solution, but rather to many different, depending on what we choose "x" to be. This also means we can draw a graph from the ifinite amount of values we can get out our function $f \left(x\right) = {x}^{2} + 6$.

Next, choose some values for "x" and connect the results together with lines on a coordinate system. Alternatively, if you have access to any kind of graphing tool, use that instead.

Next, you have two options.
Either look at your graph and find the value/values for "x", where "y" is 0. This works because you know that you are looking "y" when it is 0.

Alternatively, you can define a new function from the right side of the equation. Such a function could be $g \left(x\right) = 0$.
Use your graphing tool to create a graph of this function too, then find where these graphs intersect, then write down the "x" value/values.
This method is more sustainable, as you might not always know exactly what "y" value to look for.

Now if you have done all that, you might have noticed something strange. The two graphs doesn't intersect! This means that there is no particular solution for ${x}^{2} + 6 = 0$. That is unless you use complex numbers, but that's probably for another day :)

Here is the graph of $f \left(x\right)$. graph{x^2+6 [-17.74, 22.8, -1.55, 18.73]} graph{0 [-18.42, 22.12, -4.23, 16.05]}

... If you must know, using complex numbers, the solution is $x = \pm \sqrt{6} i$ :)