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If-x-2-x-3-0-Find-x-9-x-2-




Question Number 212448 by hardmath last updated on 13/Oct/24
If  x^2  − x + 3 = 0  Find  x + (9/x^2 ) = ?
$$\mathrm{If}\:\:\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{x}\:+\:\mathrm{3}\:=\:\mathrm{0} \\ $$$$\mathrm{Find}\:\:\mathrm{x}\:+\:\frac{\mathrm{9}}{\mathrm{x}^{\mathrm{2}} }\:=\:? \\ $$
Answered by A5T last updated on 13/Oct/24
x^2 −x+3=0⇒x^3 =x^2 −3x=x−3−3x=−2x−3  x+(9/x^2 )=((x^3 +9)/x^2 )=((−2x−3+9)/(x−3))=((−2(x−3))/(x−3))=−2
$${x}^{\mathrm{2}} −{x}+\mathrm{3}=\mathrm{0}\Rightarrow{x}^{\mathrm{3}} ={x}^{\mathrm{2}} −\mathrm{3}{x}={x}−\mathrm{3}−\mathrm{3}{x}=−\mathrm{2}{x}−\mathrm{3} \\ $$$${x}+\frac{\mathrm{9}}{{x}^{\mathrm{2}} }=\frac{{x}^{\mathrm{3}} +\mathrm{9}}{{x}^{\mathrm{2}} }=\frac{−\mathrm{2}{x}−\mathrm{3}+\mathrm{9}}{{x}−\mathrm{3}}=\frac{−\mathrm{2}\left({x}−\mathrm{3}\right)}{{x}−\mathrm{3}}=−\mathrm{2} \\ $$
Answered by A5T last updated on 13/Oct/24
x^2 −x+3=0⇒^(/x^2 ) 1−(1/x)+(3/x^2 )=0⇒(9/x^2 )=−3+(3/x)...(i)  x^2 −x+3=0⇒^(/x) x−1+(3/x)=0⇒x=1−(3/x)...(ii)  (i)+(ii)⇒(9/x^2 )+x=−2
$${x}^{\mathrm{2}} −{x}+\mathrm{3}=\mathrm{0}\overset{/{x}^{\mathrm{2}} } {\Rightarrow}\mathrm{1}−\frac{\mathrm{1}}{{x}}+\frac{\mathrm{3}}{{x}^{\mathrm{2}} }=\mathrm{0}\Rightarrow\frac{\mathrm{9}}{{x}^{\mathrm{2}} }=−\mathrm{3}+\frac{\mathrm{3}}{{x}}…\left({i}\right) \\ $$$${x}^{\mathrm{2}} −{x}+\mathrm{3}=\mathrm{0}\overset{/{x}} {\Rightarrow}{x}−\mathrm{1}+\frac{\mathrm{3}}{{x}}=\mathrm{0}\Rightarrow{x}=\mathrm{1}−\frac{\mathrm{3}}{{x}}…\left({ii}\right) \\ $$$$\left({i}\right)+\left({ii}\right)\Rightarrow\frac{\mathrm{9}}{{x}^{\mathrm{2}} }+{x}=−\mathrm{2} \\ $$
Answered by Ghisom last updated on 14/Oct/24
3=x(1−x)  9=x^2 (1−x)^2   x+((x^2 (1−x)^2 )/x^2 )=x^2 −x+1=−2
$$\mathrm{3}={x}\left(\mathrm{1}−{x}\right) \\ $$$$\mathrm{9}={x}^{\mathrm{2}} \left(\mathrm{1}−{x}\right)^{\mathrm{2}} \\ $$$${x}+\frac{{x}^{\mathrm{2}} \left(\mathrm{1}−{x}\right)^{\mathrm{2}} }{{x}^{\mathrm{2}} }={x}^{\mathrm{2}} −{x}+\mathrm{1}=−\mathrm{2} \\ $$

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