Question

Find all real solutions of the equation, rounded to two decimals. x^{4}-8x^{2}+2=0

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ANSWERED
asked 2021-06-11
Find all real solutions of the equation, rounded to two decimals.
\(x^{4}-8x^{2}+2=0\)

Answers (1)

2021-06-12

Step 1
To find: all real solutions of the equation, rounded to two decimals.
\(x^{4}-8x^{2}+2=0\)
Let \(x^{2}=y\)

Step 2
\(x^{4}-8x^{2}+2=0\)
if \(x^{2}=y\), then
\(y^{2}-8y+2=0\)
\(\Rightarrow(y-4)^{2}-16+2=0\)
\(\Rightarrow(y-4)^{2}=14\)
\(\Rightarrow\ y\pm\sqrt{14}\)
therefore, \(y=x^{2}=4\pm\sqrt{14}\) \(\Rightarrow\begin{array}{|c|}\hline x=\pm\sqrt{(4\pm\sqrt{14})} \\ \hline \end{array}\) are the real solutions
and the solutions are \(x=\sqrt{4+\sqrt{14}}=2.78,\)
\(x=\sqrt{4-\sqrt{14}}=0.51,\)
\(x=-\sqrt{4+\sqrt{14}}=-2.78,\) and
\(x=-\sqrt{4-\sqrt{14}}=0.51\)

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