Quadratic EquationHard
Question
Consider the equation $3x^{7} - x^{4} - 30x^{5} + 10x^{2} + 3x^{3} - 1 = 0$ then
Options
A.The minimum real root of equation is $( - \sqrt{3} - \sqrt{2})$.
B.The minimum real root of equation is $( - \sqrt{3} + \sqrt{2})$.
C.The maximum real root of equation is $(\sqrt{3} + \sqrt{2})$.
D.The number of positive roots of equation is 3 .
Solution
$3x^{3}\left( x^{4} - 10x^{2} + 1 \right) - \left( x^{4} - 10x^{2} + 1 \right) = 0$
$$\begin{matrix} \left( 3x^{3} - 1 \right)\left( x^{4} - 10x^{2} + 1 \right) = 0 \Rightarrow x = \frac{1}{\sqrt[3]{3}} \\ x^{2} = 5 \pm 2\sqrt{6} = (\sqrt{3} \pm \sqrt{2})^{2} \\ x = \sqrt{3} + \sqrt{2}, - \sqrt{3} - \sqrt{2},\sqrt{3} - \sqrt{2}, - \sqrt{3} + \sqrt{2},\frac{1}{\sqrt[3]{3}} \end{matrix}$$
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