Question-91473

Question Number 91473 by A8;15: last updated on 01/May/20

Answered by MJS last updated on 01/May/20

x^4 −22x^2 +x+112=0 first, testing all factors of ±112=±2^4 7 ⇒ no solution second, trying to find the square factors (x^2 −αx−β)(x^2 +αx−γ)=x^4 −22x^2 +x+112 by matching the constants ⇒ no “nice” solution in this case now you can try Ferrari′s solution but it will not be useable ⇒ approximate x_1 ≈−3.81841 x_2 ≈−2.75171 x_3 ≈2.92139 x_4 ≈3.64874

$${x}^{\mathrm{4}} −\mathrm{22}{x}^{\mathrm{2}} +{x}+\mathrm{112}=\mathrm{0} \\ $$$$\mathrm{first},\:\mathrm{testing}\:\mathrm{all}\:\mathrm{factors}\:\mathrm{of}\:\pm\mathrm{112}=\pm\mathrm{2}^{\mathrm{4}} \mathrm{7} \\ $$$$\Rightarrow\:\mathrm{no}\:\mathrm{solution} \\ $$$$\mathrm{second},\:\mathrm{trying}\:\mathrm{to}\:\mathrm{find}\:\mathrm{the}\:\mathrm{square}\:\mathrm{factors} \\ $$$$\left({x}^{\mathrm{2}} −\alpha{x}−\beta\right)\left({x}^{\mathrm{2}} +\alpha{x}−\gamma\right)={x}^{\mathrm{4}} −\mathrm{22}{x}^{\mathrm{2}} +{x}+\mathrm{112} \\ $$$$\mathrm{by}\:\mathrm{matching}\:\mathrm{the}\:\mathrm{constants} \\ $$$$\Rightarrow\:\mathrm{no}\:“\mathrm{nice}''\:\mathrm{solution}\:\mathrm{in}\:\mathrm{this}\:\mathrm{case} \\ $$$$\mathrm{now}\:\mathrm{you}\:\mathrm{can}\:\mathrm{try}\:\mathrm{Ferrari}'\mathrm{s}\:\mathrm{solution}\:\mathrm{but}\:\mathrm{it}\:\mathrm{will} \\ $$$$\mathrm{not}\:\mathrm{be}\:\mathrm{useable} \\ $$$$\Rightarrow\:\mathrm{approximate} \\ $$$${x}_{\mathrm{1}} \approx−\mathrm{3}.\mathrm{81841} \\ $$$${x}_{\mathrm{2}} \approx−\mathrm{2}.\mathrm{75171} \\ $$$${x}_{\mathrm{3}} \approx\mathrm{2}.\mathrm{92139} \\ $$$${x}_{\mathrm{4}} \approx\mathrm{3}.\mathrm{64874} \\ $$

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