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Question-148428




Question Number 148428 by 0731619 last updated on 27/Jul/21
Answered by Mathspace last updated on 27/Jul/21
(((a+b)^2 )/(ab))−3 =(((a+b)^2 −3ab)/(ab))  =((a^2 +2ab+b^2 −3ab)/(ab))  =((a^2 −ab+b^2 )/(ab))=((a^2 −2a(b/2)+(b^2 /4)+b^2 −(b^2 /4))/(ab))  =(((a−(b/(2 )))^2 +((3b^2 )/4))/(ab))>0
$$\frac{\left({a}+{b}\right)^{\mathrm{2}} }{{ab}}−\mathrm{3}\:=\frac{\left({a}+{b}\right)^{\mathrm{2}} −\mathrm{3}{ab}}{{ab}} \\ $$$$=\frac{{a}^{\mathrm{2}} +\mathrm{2}{ab}+{b}^{\mathrm{2}} −\mathrm{3}{ab}}{{ab}} \\ $$$$=\frac{{a}^{\mathrm{2}} −{ab}+{b}^{\mathrm{2}} }{{ab}}=\frac{{a}^{\mathrm{2}} −\mathrm{2}{a}\frac{{b}}{\mathrm{2}}+\frac{{b}^{\mathrm{2}} }{\mathrm{4}}+{b}^{\mathrm{2}} −\frac{{b}^{\mathrm{2}} }{\mathrm{4}}}{{ab}} \\ $$$$=\frac{\left({a}−\frac{{b}}{\mathrm{2}\:}\right)^{\mathrm{2}} +\frac{\mathrm{3}{b}^{\mathrm{2}} }{\mathrm{4}}}{{ab}}>\mathrm{0} \\ $$$$ \\ $$

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