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Category: Algebra

Find-the-sum-of-the-solutions-of-the-equation-x-2-x-x-4-2-0-x-gt-0-

Question Number 182073 by Acem last updated on 04/Dec/22 $${Find}\:{the}\:{sum}\:{of}\:{the}\:{solutions}\:{of}\:{the}\:{equation}: \\ $$$$\:\mid\sqrt{{x}}\:−\:\mathrm{2}\mid+\:\sqrt{{x}}\:\left(\sqrt{{x}}\:−\:\mathrm{4}\right)+\:\mathrm{2}=\:\mathrm{0}\:\:\:;\:{x}>\:\mathrm{0} \\ $$ Answered by HeferH last updated on 04/Dec/22 $$\:\mid\sqrt{{x}}\:−\:\mathrm{2}\mid\:=\:\mathrm{4}\sqrt{{x}}\:−\:{x}\:−\:\mathrm{2} \\ $$$$\:\:\sqrt{{x}}\:−\:\mathrm{2}\:=\:\pm\:\left(\mathrm{4}\sqrt{{x}}\:−\:{x}\:−\:\mathrm{2}\right) \\…

Let-x-xy-y-54-x-y-N-Find-x-y-

Question Number 182074 by Acem last updated on 04/Dec/22 $${Let}\:{x}+\:{xy}+\:{y}=\:\mathrm{54}\:\:\:;\:{x},\:{y}\in\:\mathbb{N}\:,\:{Find}\:{x}+\:{y} \\ $$ Answered by HeferH last updated on 04/Dec/22 $${x}\:+\:{xy}\:+\:{y}\:=\:\mathrm{54} \\ $$$$\:{x}\left(\mathrm{1}\:+\:{y}\right)\:+\:{y}\:=\:\mathrm{54} \\ $$$$\:{x}\left(\mathrm{1}\:+\:{y}\right)\:+\:\left(\mathrm{1}\:+\:{y}\right)\centerdot\mathrm{1}\:=\:\mathrm{55} \\…

Question-116507

Question Number 116507 by bemath last updated on 04/Oct/20 Answered by bobhans last updated on 04/Oct/20 $$\mathrm{let}\:\frac{\mathrm{1}}{\frac{\mathrm{1}}{\mathrm{x}}+\frac{\mathrm{1}}{\mathrm{2}}}\:=\:\mathrm{r}\:\Rightarrow\:\frac{\mathrm{1}}{\frac{\mathrm{2}+\mathrm{x}}{\mathrm{2x}}}=\:\mathrm{r} \\ $$$$\Rightarrow\frac{\mathrm{2x}}{\mathrm{2}+\mathrm{x}}\:=\:\mathrm{r}\:.\:\mathrm{the}\:\mathrm{equation}\:\mathrm{equivalent} \\ $$$$\mathrm{to}\:\frac{\mathrm{1}}{\left[\frac{\mathrm{1}}{\mathrm{r}+\mathrm{r}}+\frac{\mathrm{1}}{\mathrm{r}+\mathrm{r}}\right]}\:=\:\frac{\mathrm{x}}{\mathrm{36}}\:\Rightarrow\frac{\mathrm{1}}{\left(\frac{\mathrm{1}}{\mathrm{2r}}+\frac{\mathrm{1}}{\mathrm{2r}}\right)}\:=\:\frac{\mathrm{x}}{\mathrm{36}} \\ $$$$\Rightarrow\:\frac{\mathrm{1}}{\left(\frac{\mathrm{1}}{\mathrm{r}}\right)}\:=\:\frac{\mathrm{x}}{\mathrm{36}}\:;\:\mathrm{r}\:=\:\frac{\mathrm{x}}{\mathrm{36}} \\ $$$$\Rightarrow\frac{\mathrm{2x}}{\mathrm{2}+\mathrm{x}}\:=\:\frac{\mathrm{x}}{\mathrm{36}}\:;\:\mathrm{x}\neq\mathrm{0}…

Question-50972

Question Number 50972 by Necxx last updated on 22/Dec/18 Answered by mr W last updated on 22/Dec/18 $$\mathrm{2}^{\mathrm{2}{x}} =\mathrm{8}{x} \\ $$$${e}^{\mathrm{2}{x}\mathrm{ln}\:\mathrm{2}} =\mathrm{8}{x} \\ $$$$\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{4}}{e}^{\mathrm{2}{x}\mathrm{ln}\:\mathrm{2}} =\mathrm{2}{x}\mathrm{ln}\:\mathrm{2}…