Question Number 149060 by mathdanisur last updated on 02/Aug/21
$${if}\:\:\:\mathrm{4}{z}\sqrt{{z}}\:−\:\mathrm{11}\sqrt{{z}}\:=\:\mathrm{5} \\ $$$${find}\:\:\:\mathrm{2}{z}\:−\:\sqrt{{z}}\:=\:? \\ $$
Answered by bramlexs22 last updated on 02/Aug/21
$$\Rightarrow\sqrt{\mathrm{z}}\:=\:\mathrm{u}\: \\ $$$$\Rightarrow\mathrm{4z}\sqrt{\mathrm{z}}−\sqrt{\mathrm{z}}\:=\:\mathrm{5}+\mathrm{10}\sqrt{\mathrm{z}} \\ $$$$\Rightarrow\sqrt{\mathrm{z}}\:\left(\mathrm{4z}−\mathrm{1}\right)=\mathrm{5}\left(\mathrm{1}+\mathrm{2}\sqrt{\mathrm{z}}\:\right) \\ $$$$\Rightarrow\sqrt{\mathrm{z}}\:\left(\mathrm{2}\sqrt{\mathrm{z}}−\mathrm{1}\right)\left(\mathrm{2}\sqrt{\mathrm{z}}+\mathrm{1}\right)=\mathrm{5}\left(\mathrm{2}\sqrt{\mathrm{z}}+\mathrm{1}\right) \\ $$$$\Rightarrow\left(\mathrm{2}\sqrt{\mathrm{z}}+\mathrm{1}\right)\left\{\sqrt{\mathrm{z}}\:\left(\mathrm{2}\sqrt{\mathrm{z}}−\mathrm{1}\right)−\mathrm{5}\right\}=\mathrm{0} \\ $$$$\mathrm{2}\sqrt{\mathrm{z}}\:+\mathrm{1}\:=\:\mathrm{0}\:\leftleftarrows\:\mathrm{not}\:\mathrm{possible} \\ $$$$\sqrt{\mathrm{z}}\:\left(\mathrm{2}\sqrt{\mathrm{z}}−\mathrm{1}\right)−\mathrm{5}=\mathrm{0} \\ $$$$\Rightarrow\mathrm{2z}−\sqrt{\mathrm{z}}\:=\:\sqrt{\mathrm{z}}\:\left(\mathrm{2}\sqrt{\mathrm{z}}−\mathrm{1}\right)=\:\mathrm{5} \\ $$$$ \\ $$
Commented by mathdanisur last updated on 02/Aug/21
$${Thank}\:{You}\:{Ser} \\ $$