Find-x-gt-1-a-b-x-1-a-1-b-1-x-1-a-b-x-1-x-1-a-1-b-x-a-b-x-x-a-b-x-b-a-ab-x-a-b-x-ax-bx-x-2-b-a-ab-a-b-x-2-ax-bx-

Question Number 227898 by needothink last updated on 26/Feb/26

Find x = ? => (1/(a+b+x)) = (1/a) + (1/b) + (1/x) = (1/(a+b+x)) −(1/x) = (1/a) + (1/b) = ((x −(a+b+x))/(x(a+b+x))) = ((b+a)/(ab)) = ((x−a−b−x)/(ax+bx+x^2 )) = ((b+a)/(ab)) = ((−a−b )/(x^2 +ax+bx )) = ((b+a)/(ab)) = ((−(a+b) )/(x^2 +ax+bx )) = ((b+a)/(ab)) = ((−1 )/(x^2 +ax+bx )) = (1/(ab)) = −ab = x^2 +ax+bx = 0= x^2 +ax+bx+ab = 0 = x(x+a)+b(x+a) = 0 = (x+a)(x+b) => x+a = 0 => x+b = 0 x = −a x = −b

$$\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{x}}\:=\:? \\ $$$$=>\:\frac{\mathrm{1}}{\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{b}}+\boldsymbol{\mathrm{x}}}\:=\:\frac{\mathrm{1}}{\boldsymbol{\mathrm{a}}}\:+\:\frac{\mathrm{1}}{\boldsymbol{\mathrm{b}}}\:+\:\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}} \\ $$$$=\:\frac{\mathrm{1}}{\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{b}}+\boldsymbol{\mathrm{x}}}\:−\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}}\:=\:\frac{\mathrm{1}}{\boldsymbol{\mathrm{a}}}\:+\:\frac{\mathrm{1}}{\boldsymbol{\mathrm{b}}} \\ $$$$=\:\frac{\boldsymbol{\mathrm{x}}\:−\left(\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{b}}+\boldsymbol{\mathrm{x}}\right)}{\boldsymbol{\mathrm{x}}\left(\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{b}}+\boldsymbol{\mathrm{x}}\right)}\:=\:\frac{\boldsymbol{\mathrm{b}}+\boldsymbol{\mathrm{a}}}{\boldsymbol{\mathrm{ab}}} \\ $$$$=\:\frac{\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{a}}−\boldsymbol{\mathrm{b}}−\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{ax}}+\boldsymbol{\mathrm{bx}}+\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\:=\:\frac{\boldsymbol{\mathrm{b}}+\boldsymbol{\mathrm{a}}}{\boldsymbol{\mathrm{ab}}} \\ $$$$=\:\frac{−\boldsymbol{\mathrm{a}}−\boldsymbol{\mathrm{b}}\:}{\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{ax}}+\boldsymbol{\mathrm{bx}}\:}\:=\:\frac{\boldsymbol{\mathrm{b}}+\boldsymbol{\mathrm{a}}}{\boldsymbol{\mathrm{ab}}} \\ $$$$=\:\frac{−\left(\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{b}}\right)\:}{\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{ax}}+\boldsymbol{\mathrm{bx}}\:}\:=\:\frac{\boldsymbol{\mathrm{b}}+\boldsymbol{\mathrm{a}}}{\boldsymbol{\mathrm{ab}}} \\ $$$$=\:\frac{−\mathrm{1}\:}{\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{ax}}+\boldsymbol{\mathrm{bx}}\:}\:=\:\frac{\mathrm{1}}{\boldsymbol{\mathrm{ab}}} \\ $$$$=\:−\boldsymbol{\mathrm{ab}}\:=\:\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{ax}}+\boldsymbol{\mathrm{bx}} \\ $$$$=\:\mathrm{0}=\:\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{ax}}+\boldsymbol{\mathrm{bx}}+\boldsymbol{\mathrm{ab}} \\ $$$$=\:\mathrm{0}\:=\:\boldsymbol{\mathrm{x}}\left(\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{a}}\right)+\boldsymbol{\mathrm{b}}\left(\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{a}}\right) \\ $$$$=\:\mathrm{0}\:=\:\left(\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{a}}\right)\left(\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{b}}\right) \\ $$$$=>\:\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{a}}\:=\:\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=>\:\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{b}}\:=\:\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{x}}\:=\:−\boldsymbol{\mathrm{a}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{x}}\:=\:−\boldsymbol{\mathrm{b}} \\ $$

Commented by AgniMath last updated on 01/Mar/26