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Question Number 204461 by MATHEMATICSAM last updated on 18/Feb/24

$$\mathrm{Solve}\mathrm{for}x$$$$(x-12)(x-13)=\frac{34}{{33}^2}$$

Answered by TonyCWX08 last updated on 18/Feb/24

$$x^2-25x+156=\frac{34}{1089}$$$$x^2-25=\frac{34}{1089}-156$$$$x^2-25+{\left(\frac{25}{2}\right)}^2=\frac{-164859}{1057}+\frac{625}{4}$$$${(x-\frac{25}{2})}^2=\frac{304}{1081}$$$$x-\frac{25}{2}=\pm \sqrt{\frac{304}{1081}}$$$$x=\frac{25}{2}\pm \sqrt{\frac{304}{1081}}$$$$x_1=\frac{395}{33}{x}_2=\frac{430}{33}$$

Answered by som(math1967) last updated on 18/Feb/24

$$let33=a\Rightarrow 34=a+1$$$$(x-12)(x-13)=\frac{a+1}{a^2}$$$$\Rightarrow a^2(x-12)(x-13)-a-1=0$$$$\Rightarrow a^2(x-12)(x-13)+(x-13)a$$$$-(x-12)a-1=0$$$$\Rightarrow a(x-13)(ax-12a+1)$$$$-1(ax-12a+1)=0$$$$\Rightarrow (ax-12a+1)(ax-13a-1)=0$$$$\therefore x=\frac{12a-1}{a}=12-\frac{1}{a}=12-\frac{1}{33}$$$$x=11\frac{32}{33}$$$$orx=\frac{13a+1}{a}=13+\frac{1}{a}=13\frac{1}{33}$$

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