Question

64(9)^{x} -84(12)^{x} +27(16)^{x} =0
find x

Answer 1

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$given \: = > \\ \\ 64( {9}^{x} ) - 84( {12}^{x} ) + 27( {16}^{x} ) = 0 \\ \\ = > 64( {3}^{2x} ) - 84( {3}^{x} . {4}^{x} ) + 27( {4}^{2x} ) = 0 \\ \\ dividing \: whole \: equation \: by \: ( {4}^{2x} ) \: \: \: \\ \\ we \: get \: = > \\ \\ = > 64 (\frac{ {3}^{2x} }{ {4}^{2x} } ) - 84( \frac{ {3}^{x} . {4}^{x} }{ {4}^{2x} } ) + 27( \frac{ {4}^{2x} }{ {4}^{2x} } ) = 0 \\ \\ = > 64( { \frac{3}{4}) }^{2x} - 84( { \frac{3}{4} )}^{x} + 27 = 0 \\ \\ lets \:( { \frac{3}{4} )}^{x} = m \\ \\ = > 64 {m}^{2} - 84m + 27 = 0 \\ \\ using \: quadratic \: formula \: we \: get \\ \\ = > m = \frac{ - b \binom{ + }{ - } \sqrt{ {b}^{2} - 4ac } }{2a} \\ \\ = > m = \frac{ - ( - 84) \binom{ + }{ - } \sqrt{ {84}^{2} - 4 \times 64 \times 27 } }{2 \times 64} \\ \\ = > m = \frac{84 \binom{ + }{ - } \sqrt{7056 - 6912} }{128} \\ \\ = > m = \frac{84 \binom{ + }{ - } \sqrt{144} }{128} \\ \\ = > m = \frac{84 \binom{ + }{ - }12 }{128} \\ \\ = > m = \frac{84 + 12}{128} \: \: \: \: or \: \: \: \: m = \frac{84 - 12}{128} \\ \\ = > m = \frac{3}{4} \: \: \: or \: \: \: \: m = \frac{9}{16} \\ \\ now \\ \\ { (\frac{3}{4}) }^{x} = \frac{3}{4} \: \: \: \: \: or \: \: \: \: \: ( { \frac{3}{4} )}^{x} = \frac{9}{16} \\ \\ { (\frac{3}{4}) }^{x} = { (\frac{3}{4} )}^{1} \: \: \: \: or \: \: \: \: \: ( { \frac{3}{4}) }^{x} = { (\frac{3}{4}) }^{2} \\ \\ comparing \: both \: sides \: we \: get \\ \\ = > \: \: \: x = 1 \: \: or \: \: \: x = 2 \: \: \: \: .....answer$


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