<PRE><B>Question 1184166</B>
&lt;br&gt;
{{{ f(a) = a^6 -12a^4 + 48a^2 - 61 }}}&lt;br&gt;
Graphing the function will help us see if our results are reasonable.  Here is a graph:&lt;br&gt;
{{{graph(400,400,-3,3,-20,20,x^6-12x^4+48x^2-61)}}}&lt;br&gt;
Note the graph is even (symmetrical with respect to the y-axis, because every exponent in the polynomial is even.&lt;br&gt;
There is no general method for solving a 6th degree polynomial equation.&lt;br&gt;
By far the easiest way to find the APPROXIMATE real roots is by graphing the function on a graphing calculator and finding the zeros from the graph.&lt;br&gt;
My TI-83 shows the real roots to be +/- 1.599296855 to 9 decimal places, which is consistent with the graph.&lt;br&gt;
You can get that same result using Newton&#39;s Method.  Look on the internet for a video on how to use Newton&#39;s Method on a TI83 or TI84 calculator.&lt;br&gt;
I would consider doing that very worthwhile, since Newton&#39;s Method is a very powerful tool.&lt;br&gt;
Finally, there is a way to find the exact real roots for this example.&lt;br&gt;
Since the function is even,  let y=a^2 and write the equation as&lt;br&gt;
x^3-12x^2+48x-61=0&lt;br&gt;
Then look again on the internet and find the formula for the general solution of a cubic equation and use it; then remember that you are solving for a=sqrt(x), not for x.&lt;br&gt;
It is a very ugly formula, which I have never tried to use; and I have no inclination to try it on your example.  So if you want to find the exact real zeros, I say have fun...!&lt;br&gt;
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