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Question Number 167772 by mathlove last updated on 24/Mar/22

x+(√x)=5  x+(5/( (√x)))=?

$${x}+\sqrt{{x}}=\mathrm{5} \\ $$$${x}+\frac{\mathrm{5}}{\:\sqrt{{x}}}=? \\ $$

Commented by mr W last updated on 24/Mar/22

((√x))^2 +(√x)−5=0  (√x)=(((√(21))−1)/2)  x+(5/( (√x)))=5+(5/( (√x)))−(√x)  =5+((10)/( (√(21))−1))−(((√(21))−1)/2)  =5+(((√(21))+1)/( 2))−(((√(21))−1)/2)  =6

$$\left(\sqrt{{x}}\right)^{\mathrm{2}} +\sqrt{{x}}−\mathrm{5}=\mathrm{0} \\ $$$$\sqrt{{x}}=\frac{\sqrt{\mathrm{21}}−\mathrm{1}}{\mathrm{2}} \\ $$$${x}+\frac{\mathrm{5}}{\:\sqrt{{x}}}=\mathrm{5}+\frac{\mathrm{5}}{\:\sqrt{{x}}}−\sqrt{{x}} \\ $$$$=\mathrm{5}+\frac{\mathrm{10}}{\:\sqrt{\mathrm{21}}−\mathrm{1}}−\frac{\sqrt{\mathrm{21}}−\mathrm{1}}{\mathrm{2}} \\ $$$$=\mathrm{5}+\frac{\sqrt{\mathrm{21}}+\mathrm{1}}{\:\mathrm{2}}−\frac{\sqrt{\mathrm{21}}−\mathrm{1}}{\mathrm{2}} \\ $$$$=\mathrm{6} \\ $$

Commented by MJS_new last updated on 24/Mar/22

x+(√x)=a ⇒ x+(a/( (√x)))=a+1  prove:  (1)  a=x+(√x)=((√x)+1)(√x)=(((1+(√x))(1−(√x))(√x))/(1−(√x)))=  =(((1−x)(√x))/(1−(√x)))  (2)  a=(((1−x)(√x))/(1−(√x)))  a(1−(√x))=(1−x)(√x)  a−a(√x)=(1−x)(√x)  (a/( (√x)))−a=1−x  x−(a/( (√x)))=a+1

$${x}+\sqrt{{x}}={a}\:\Rightarrow\:{x}+\frac{{a}}{\:\sqrt{{x}}}={a}+\mathrm{1} \\ $$$$\mathrm{prove}: \\ $$$$\left(\mathrm{1}\right) \\ $$$${a}={x}+\sqrt{{x}}=\left(\sqrt{{x}}+\mathrm{1}\right)\sqrt{{x}}=\frac{\left(\mathrm{1}+\sqrt{{x}}\right)\left(\mathrm{1}−\sqrt{{x}}\right)\sqrt{{x}}}{\mathrm{1}−\sqrt{{x}}}= \\ $$$$=\frac{\left(\mathrm{1}−{x}\right)\sqrt{{x}}}{\mathrm{1}−\sqrt{{x}}} \\ $$$$\left(\mathrm{2}\right) \\ $$$${a}=\frac{\left(\mathrm{1}−{x}\right)\sqrt{{x}}}{\mathrm{1}−\sqrt{{x}}} \\ $$$${a}\left(\mathrm{1}−\sqrt{{x}}\right)=\left(\mathrm{1}−{x}\right)\sqrt{{x}} \\ $$$${a}−{a}\sqrt{{x}}=\left(\mathrm{1}−{x}\right)\sqrt{{x}} \\ $$$$\frac{{a}}{\:\sqrt{{x}}}−{a}=\mathrm{1}−{x} \\ $$$${x}−\frac{{a}}{\:\sqrt{{x}}}={a}+\mathrm{1} \\ $$

Answered by mnjuly1970 last updated on 25/Mar/22

  x + ((x+(√x))/( (√x)))= x+(√x) +1= 6

$$\:\:{x}\:+\:\frac{{x}+\sqrt{{x}}}{\:\sqrt{{x}}}=\:{x}+\sqrt{{x}}\:+\mathrm{1}=\:\mathrm{6} \\ $$

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