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Answer 1

3.

$\rm \frac{1}{x - 1} + \frac{2}{x - 2} = \frac{6}{x - 3} \\ \\ \to \tt \frac{(x - 2) + 2(x - 1)}{(x - 1)(x - 2)} = \frac{6}{x - 3} \\ \\ \to \tt \frac{x - 2 + 2x - 2}{ {x}^{2} - x - 2x + 2} = \frac{6}{x - 3} \\ \\ \to \tt \frac{ 3x - 4 }{ {x}^{2} - 3x + 2 } = \frac{6}{x - 3} \\ \\ \to \tt(3x - 4)(x - 3) = (6)( {x}^{2} - 3x + 2 ) \\ \to \tt {3x}^{2} - 4x - 3x + 12 = {6x}^{2} - 18x + 12 \\ \to \tt {3x}^{2} - 7x + \cancel{12} = {6x}^{2} - 18x + \cancel{12} \\ \to \tt {3x}^{2} - {6x}^{2} - 7x + 18x = 0 \\ \to \tt - {3x}^{2} + 11x = 0 \\ \to \tt( - )( { - 3x}^{2} + 11x ) = ( - 1)0 \\ \to \tt{3x}^{2} - 11x = 0 \\\to \tt x(3x - 11 ) = 0 \\ \\ \huge\therefore \: \: \: \sf x = \red 0 \: \: \: \rm{or} \: \: \sf \red{\frac{11}{3} }$

4.

$\large \rm {2x}^{2} + 21x - 50 = 0 \\ \\ \tt \to {( \sqrt{2} x)}^{2} + 2( \sqrt{2}x ) \bigg( \frac{21}{2 \sqrt{2} } \bigg) - 50 + {\bigg( \frac{21}{2 \sqrt{2} } \bigg)}^{2} - {\bigg( \frac{21}{2 \sqrt{2} } \bigg)}^{2} = 0 \\ \\ \to\tt {\bigg\{ \sqrt{2}x + \frac{21}{2 \sqrt{2} } \bigg\}}^{2} - 50 - \frac{441}{8} = 0 \\ \\ \to\tt {\bigg\{ \sqrt{2}x + \frac{21}{2 \sqrt{2} } \bigg\}}^{2} - \bigg\{50 + \frac{441}{8} \bigg\} = 0 \\ \\ \to\tt{\bigg\{ \sqrt{2}x + \frac{21}{2 \sqrt{2} } \bigg\}}^{2} - \bigg\{ \frac{400 + 441}{8} \bigg\} = 0 \\ \\ \to\tt{\bigg\{ \sqrt{2}x + \frac{21}{2 \sqrt{2} } \bigg\}}^{2} - \frac{841}{8} = 0 \\ \\ \to\tt{\bigg\{ \sqrt{2}x + \frac{21}{2 \sqrt{2} } \bigg\}}^{2} - {\bigg\{ \sqrt{ \frac{841}{8} } \bigg\}}^{2} = 0 \\ \\ \tt \to \bigg\{ \sqrt{2} x + \frac{21}{2 \sqrt{2} } + \sqrt{ \frac{841}{8} } \bigg\} \bigg\{ \sqrt{2} x + \frac{21}{2 \sqrt{2} } - \sqrt{ \frac{841}{8} } \bigg\} = 0 \\ \\ \tt \to \bigg\{ \sqrt{2} x + \frac{21}{2 \sqrt{2} } + \frac{ \sqrt{841} }{2 \sqrt{2} } \bigg\}\bigg\{ \sqrt{2} x + \frac{21}{2 \sqrt{2} } - \frac{ \sqrt{841} }{2 \sqrt{2} } \bigg\} = 0 \\ \\ \tt \to \bigg \{ \frac{4x + 21 + \sqrt{841} }{2 \sqrt{2} } \bigg\}\bigg \{ \frac{4x + 21 - \sqrt{841} }{2 \sqrt{2} } \bigg \} = 0 \\ \\ \tt \to \frac{1}{2 \sqrt{2} } \bigg \{ 4x + 21 + \sqrt{841} \bigg \} \bigg \{ 4x + 21 - \sqrt{841} \bigg \} = 0 \\ \\ \tt \to \bigg \{ 4x + 21 + \sqrt{841} \bigg \} \bigg \{ 4x + 21 - \sqrt{841} \bigg \} = \frac{0}{ \frac{1}{2 \sqrt{2} } } \\ \\ \tt \to \bigg \{ \red{4x + 21 + \sqrt{841}} \bigg \} \bigg \{ \red{4x + 21 - \sqrt{841} }\bigg \} = 0$

Answer 2

Answer:

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