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7x-5-4x-4-9x-3-12x-2-5x-9-0-How-many-roots-of-this-equation-are-Negative-




Question Number 13438 by Nayon last updated on 20/May/17
7x^5 −4x^4 +9x^3 +12x^2 +5x−9=0  How many roots of this equation  are Negative?
$$\mathrm{7}{x}^{\mathrm{5}} −\mathrm{4}{x}^{\mathrm{4}} +\mathrm{9}{x}^{\mathrm{3}} +\mathrm{12}{x}^{\mathrm{2}} +\mathrm{5}{x}−\mathrm{9}=\mathrm{0} \\ $$$${How}\:{many}\:{roots}\:{of}\:{this}\:{equation} \\ $$$${are}\:{Negative}? \\ $$$$ \\ $$
Answered by mrW1 last updated on 20/May/17
7x^5 −4x^4 +9x^3 +12x^2 +5x−9=0  7x^5 −9x^3 +5x−4x^4 +12x^2 −9=0  x(7x^4 −9x^2 +5)=(4x^4 −12x^2 +9)  7x(x^4 −(9/7)x^2 +(5/7))=4(x^4 −3x^2 +(9/4))  7x[x^4 −2×(9/(14))x^2 +((9/(14)))^2 +(5/7)−((9/(14)))^2 ]=4[x^4 −2×(3/2)x^2 +((3/2))^2 ]  7x[(x^2 −(9/(14)))^2 +((59)/(196))]=4(x^2 −(3/2))^2   x=((4(x^2 −(3/2))^2 )/(7[(x^2 −(9/(14)))^2 +((59)/(196))]))=((>0)/(>0))>0    ⇒there is no negative root!
$$\mathrm{7}{x}^{\mathrm{5}} −\mathrm{4}{x}^{\mathrm{4}} +\mathrm{9}{x}^{\mathrm{3}} +\mathrm{12}{x}^{\mathrm{2}} +\mathrm{5}{x}−\mathrm{9}=\mathrm{0} \\ $$$$\mathrm{7}{x}^{\mathrm{5}} −\mathrm{9}{x}^{\mathrm{3}} +\mathrm{5}{x}−\mathrm{4}{x}^{\mathrm{4}} +\mathrm{12}{x}^{\mathrm{2}} −\mathrm{9}=\mathrm{0} \\ $$$${x}\left(\mathrm{7}{x}^{\mathrm{4}} −\mathrm{9}{x}^{\mathrm{2}} +\mathrm{5}\right)=\left(\mathrm{4}{x}^{\mathrm{4}} −\mathrm{12}{x}^{\mathrm{2}} +\mathrm{9}\right) \\ $$$$\mathrm{7}{x}\left({x}^{\mathrm{4}} −\frac{\mathrm{9}}{\mathrm{7}}{x}^{\mathrm{2}} +\frac{\mathrm{5}}{\mathrm{7}}\right)=\mathrm{4}\left({x}^{\mathrm{4}} −\mathrm{3}{x}^{\mathrm{2}} +\frac{\mathrm{9}}{\mathrm{4}}\right) \\ $$$$\mathrm{7}{x}\left[{x}^{\mathrm{4}} −\mathrm{2}×\frac{\mathrm{9}}{\mathrm{14}}{x}^{\mathrm{2}} +\left(\frac{\mathrm{9}}{\mathrm{14}}\right)^{\mathrm{2}} +\frac{\mathrm{5}}{\mathrm{7}}−\left(\frac{\mathrm{9}}{\mathrm{14}}\right)^{\mathrm{2}} \right]=\mathrm{4}\left[{x}^{\mathrm{4}} −\mathrm{2}×\frac{\mathrm{3}}{\mathrm{2}}{x}^{\mathrm{2}} +\left(\frac{\mathrm{3}}{\mathrm{2}}\right)^{\mathrm{2}} \right] \\ $$$$\mathrm{7}{x}\left[\left({x}^{\mathrm{2}} −\frac{\mathrm{9}}{\mathrm{14}}\right)^{\mathrm{2}} +\frac{\mathrm{59}}{\mathrm{196}}\right]=\mathrm{4}\left({x}^{\mathrm{2}} −\frac{\mathrm{3}}{\mathrm{2}}\right)^{\mathrm{2}} \\ $$$${x}=\frac{\mathrm{4}\left({x}^{\mathrm{2}} −\frac{\mathrm{3}}{\mathrm{2}}\right)^{\mathrm{2}} }{\mathrm{7}\left[\left({x}^{\mathrm{2}} −\frac{\mathrm{9}}{\mathrm{14}}\right)^{\mathrm{2}} +\frac{\mathrm{59}}{\mathrm{196}}\right]}=\frac{>\mathrm{0}}{>\mathrm{0}}>\mathrm{0} \\ $$$$ \\ $$$$\Rightarrow{there}\:{is}\:{no}\:{negative}\:{root}! \\ $$
Commented by ajfour last updated on 20/May/17
awesome !
$${awesome}\:! \\ $$

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