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Question

# Solve by factorization and quadratic formula 2x/x-1 +2x-5/x-3=25/3

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Solution

## Dear Student, $\mathrm{I}\mathrm{assume}\mathrm{your}\mathrm{given}\mathrm{equation}\mathrm{should}\mathrm{be}:\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\frac{2\mathrm{x}}{\mathrm{x}-4}+\frac{2\mathrm{x}-5}{\mathrm{x}-3}=\frac{25}{3}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Here}\mathrm{is}\mathrm{the}\mathrm{answer}.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\frac{2\mathrm{x}}{\mathrm{x}-4}+\frac{2\mathrm{x}-5}{\mathrm{x}-3}=\frac{25}{3}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒\frac{2\mathrm{x}\left(\mathrm{x}-3\right)+\left(\mathrm{x}-4\right)\left(2\mathrm{x}-5\right)}{\left(\mathrm{x}-3\right)\left(\mathrm{x}-4\right)}=\frac{25}{3}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒\frac{2{\mathrm{x}}^{2}-6\mathrm{x}+2{\mathrm{x}}^{2}-5\mathrm{x}-8\mathrm{x}+20}{{\mathrm{x}}^{2}-7\mathrm{x}+12}=\frac{25}{3}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒\frac{4{\mathrm{x}}^{2}-19\mathrm{x}+20}{{\mathrm{x}}^{2}-7\mathrm{x}+12}=\frac{25}{3}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒25{\mathrm{x}}^{2}-175\mathrm{x}+300=12{\mathrm{x}}^{2}-57\mathrm{x}+60\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒13{\mathrm{x}}^{2}-118\mathrm{x}+240=0\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{FACTORISATION}\mathrm{METHOD}:\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒13{\mathrm{x}}^{2}-78\mathrm{x}-40\mathrm{x}+240=0\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒13\mathrm{x}\left(\mathrm{x}-6\right)-40\left(\mathrm{x}-6\right)=0\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒\left(13\mathrm{x}-40\right)\left(\mathrm{x}-6\right)=0\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒\mathrm{x}=\frac{40}{13}\mathrm{or}\mathrm{x}=6\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathbf{QUADRATIC}\mathbf{}\mathbf{FORMULA}\mathbf{}\mathbf{:}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{a}=13;\mathrm{b}=-118;\mathrm{c}=240\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Now},\mathrm{D}={\mathrm{b}}^{2}-4\mathrm{ac}={\left(-118\right)}^{2}-4\left(13\right)\left(240\right)=13924-12480=1444\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Now},\mathrm{x}=\frac{-\mathrm{b}±\sqrt{\mathrm{D}}}{2\mathrm{a}}=\frac{118±\sqrt{1444}}{26}=\frac{118±38}{26}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒\mathrm{x}=\frac{118+38}{26}\mathrm{and}\frac{118-38}{26}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒\mathrm{x}=6\mathrm{or}\mathrm{x}=\frac{40}{13}\phantom{\rule{0ex}{0ex}}$ Regards

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