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f-x-x-6-3-x-4-4x-2-find-the-zeros-




Question Number 58103 by smiak8742 last updated on 17/Apr/19
f(x)=−x^6 +3 x^4  + 4x^2 find the zeros
$${f}\left({x}\right)=−{x}^{\mathrm{6}} +\mathrm{3}\:{x}^{\mathrm{4}} \:+\:\mathrm{4}{x}^{\mathrm{2}} {find}\:{the}\:{zeros} \\ $$
Answered by MJS last updated on 17/Apr/19
x^6 −3x^4 −4x^2 =0  x^2 (x^4 −3x^2 −4)=0  x_1 =x_2 =0  x^4 −3x^2 −4=0  x=(√t)  t^2 −3t−4=0  t_1 =−1; t_2 =4  ⇒ x_3 =−i; x_4 =i; x_5 =−2; x_6 =2
$${x}^{\mathrm{6}} −\mathrm{3}{x}^{\mathrm{4}} −\mathrm{4}{x}^{\mathrm{2}} =\mathrm{0} \\ $$$${x}^{\mathrm{2}} \left({x}^{\mathrm{4}} −\mathrm{3}{x}^{\mathrm{2}} −\mathrm{4}\right)=\mathrm{0} \\ $$$${x}_{\mathrm{1}} ={x}_{\mathrm{2}} =\mathrm{0} \\ $$$${x}^{\mathrm{4}} −\mathrm{3}{x}^{\mathrm{2}} −\mathrm{4}=\mathrm{0} \\ $$$${x}=\sqrt{{t}} \\ $$$${t}^{\mathrm{2}} −\mathrm{3}{t}−\mathrm{4}=\mathrm{0} \\ $$$${t}_{\mathrm{1}} =−\mathrm{1};\:{t}_{\mathrm{2}} =\mathrm{4} \\ $$$$\Rightarrow\:{x}_{\mathrm{3}} =−\mathrm{i};\:{x}_{\mathrm{4}} =\mathrm{i};\:{x}_{\mathrm{5}} =−\mathrm{2};\:{x}_{\mathrm{6}} =\mathrm{2} \\ $$

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