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Section 1.1 : Review : Functions

14. Determine all the roots of \(W\left( x \right) = {x^4} + 6{x^2} - 27\).

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Set the function equal to zero and factor the left side as much as possible.

\[{x^4} + 6{x^2} - 27 = \left( {{x^2} - 3} \right)\left( {{x^2} + 9} \right) = 0\]

Don’t so locked into quadratic equations that the minute you see an equation that is not quadratic you decide you can’t deal with it. While this function was not a quadratic it still factored in an obvious manner.

Now, the second term will never be zero (for any real value of \(x\) anyway and in this class those tend to be the only ones we are interested in) and so we can ignore that term. The first will be zero if,

\[{x^2} - 3 = 0\hspace{0.25in} \Rightarrow \hspace{0.25in}{x^2} = 3\hspace{0.25in}\hspace{0.25in} \Rightarrow \hspace{0.25in}x = \pm \sqrt 3 \]

So, we have two real roots of this function. Note that if we allowed complex roots (which again, we aren’t really interested in for this course) there would also be two complex roots from the second term as well.