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19x-x-2-x-1-x-19-x-x-1-78-find-x-




Question Number 210906 by hardmath last updated on 21/Aug/24
((19x − x^2 )/(x + 1)) ∙ (x + ((19 − x)/(x + 1))) = 78  find:  x = ?
$$\frac{\mathrm{19x}\:−\:\mathrm{x}^{\mathrm{2}} }{\mathrm{x}\:+\:\mathrm{1}}\:\centerdot\:\left(\mathrm{x}\:+\:\frac{\mathrm{19}\:−\:\mathrm{x}}{\mathrm{x}\:+\:\mathrm{1}}\right)\:=\:\mathrm{78} \\ $$$$\mathrm{find}:\:\:\boldsymbol{\mathrm{x}}\:=\:? \\ $$
Answered by Frix last updated on 21/Aug/24
Reconstructing your equation:  x=(a/2)±((√(a^2 −4b))/2)∨x=(b/2)±((√(4a−b^2 ))/2)i     (∗)  (x^2 −ax+b)(x^2 −bx+a)=0  (x^2 −ax+b)(x^2 −bx+a)−ab(x+1)^2 =−ab(x+1)^2   x(x−(a+b))(x^2 +(a+b))=−ab(x+1)^2   ((x((a+b)−x)(x^2 +(a+b)))/((x+1)^2 ))=ab  (((a+b)x−x^2 )/(x+1))×((x^2 +(a+b))/(x+1))=ab  (((a+b)x−x^2 )/(x+1))×(x+(((a+b)−x)/(x+1)))=ab    a=13∧b=6    ((19x−x^2 )/(x+1))×(x+((19−x)/(x+1)))=78      (∗)     x=((13)/2)±((√(145))/2)∨x=3±2i
$$\mathrm{Reconstructing}\:\mathrm{your}\:\mathrm{equation}: \\ $$$${x}=\frac{{a}}{\mathrm{2}}\pm\frac{\sqrt{{a}^{\mathrm{2}} −\mathrm{4}{b}}}{\mathrm{2}}\vee{x}=\frac{{b}}{\mathrm{2}}\pm\frac{\sqrt{\mathrm{4}{a}−{b}^{\mathrm{2}} }}{\mathrm{2}}\mathrm{i}\:\:\:\:\:\left(\ast\right) \\ $$$$\left({x}^{\mathrm{2}} −{ax}+{b}\right)\left({x}^{\mathrm{2}} −{bx}+{a}\right)=\mathrm{0} \\ $$$$\left({x}^{\mathrm{2}} −{ax}+{b}\right)\left({x}^{\mathrm{2}} −{bx}+{a}\right)−{ab}\left({x}+\mathrm{1}\right)^{\mathrm{2}} =−{ab}\left({x}+\mathrm{1}\right)^{\mathrm{2}} \\ $$$${x}\left({x}−\left({a}+{b}\right)\right)\left({x}^{\mathrm{2}} +\left({a}+{b}\right)\right)=−{ab}\left({x}+\mathrm{1}\right)^{\mathrm{2}} \\ $$$$\frac{{x}\left(\left({a}+{b}\right)−{x}\right)\left({x}^{\mathrm{2}} +\left({a}+{b}\right)\right)}{\left({x}+\mathrm{1}\right)^{\mathrm{2}} }={ab} \\ $$$$\frac{\left({a}+{b}\right){x}−{x}^{\mathrm{2}} }{{x}+\mathrm{1}}×\frac{{x}^{\mathrm{2}} +\left({a}+{b}\right)}{{x}+\mathrm{1}}={ab} \\ $$$$\frac{\left({a}+{b}\right){x}−{x}^{\mathrm{2}} }{{x}+\mathrm{1}}×\left({x}+\frac{\left({a}+{b}\right)−{x}}{{x}+\mathrm{1}}\right)={ab} \\ $$$$ \\ $$$${a}=\mathrm{13}\wedge{b}=\mathrm{6} \\ $$$$ \\ $$$$\frac{\mathrm{19}{x}−{x}^{\mathrm{2}} }{{x}+\mathrm{1}}×\left({x}+\frac{\mathrm{19}−{x}}{{x}+\mathrm{1}}\right)=\mathrm{78} \\ $$$$ \\ $$$$ \\ $$$$\left(\ast\right)\:\:\:\:\:{x}=\frac{\mathrm{13}}{\mathrm{2}}\pm\frac{\sqrt{\mathrm{145}}}{\mathrm{2}}\vee{x}=\mathrm{3}\pm\mathrm{2i} \\ $$

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