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Question Number 41032 by adityasin567@gmail.com last updated on 31/Jul/18

(a/(x−a)) +(b/(x−b))=2.find x.

$$\frac{{a}}{{x}−{a}}\:+\frac{{b}}{{x}−{b}}=\mathrm{2}.{find}\:{x}. \\ $$

Answered by tanmay.chaudhury50@gmail.com last updated on 31/Jul/18

ax−ab+bx−ab=2(x^2 −ax−bx+ab)  2x^2 −2ax−2bx+2ab−ax−bx+2ab=0  2x^2 −3ax−3bx+4ab=0  2x^2 −3x(a+b)+4ab=0  x=((3(a+b)±(√(9(a+b)^2 −4.2.4ab)))/4)  =((3(a+b)±(√(9a^2 +18ab+9b^2 −32ab)))/4)  =((3(a+b)±(√(9a^2 −14ab+9b^2 )))/4)

$${ax}−{ab}+{bx}−{ab}=\mathrm{2}\left({x}^{\mathrm{2}} −{ax}−{bx}+{ab}\right) \\ $$$$\mathrm{2}{x}^{\mathrm{2}} −\mathrm{2}{ax}−\mathrm{2}{bx}+\mathrm{2}{ab}−{ax}−{bx}+\mathrm{2}{ab}=\mathrm{0} \\ $$$$\mathrm{2}{x}^{\mathrm{2}} −\mathrm{3}{ax}−\mathrm{3}{bx}+\mathrm{4}{ab}=\mathrm{0} \\ $$$$\mathrm{2}{x}^{\mathrm{2}} −\mathrm{3}{x}\left({a}+{b}\right)+\mathrm{4}{ab}=\mathrm{0} \\ $$$${x}=\frac{\mathrm{3}\left({a}+{b}\right)\pm\sqrt{\mathrm{9}\left({a}+{b}\right)^{\mathrm{2}} −\mathrm{4}.\mathrm{2}.\mathrm{4}{ab}}}{\mathrm{4}} \\ $$$$=\frac{\mathrm{3}\left({a}+{b}\right)\pm\sqrt{\mathrm{9}{a}^{\mathrm{2}} +\mathrm{18}{ab}+\mathrm{9}{b}^{\mathrm{2}} −\mathrm{32}{ab}}}{\mathrm{4}} \\ $$$$=\frac{\mathrm{3}\left({a}+{b}\right)\pm\sqrt{\mathrm{9}{a}^{\mathrm{2}} −\mathrm{14}{ab}+\mathrm{9}{b}^{\mathrm{2}} }}{\mathrm{4}} \\ $$

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