Question

X+4/x-4 - x-4/x+4 = 1 - x square / x square-16​

Answer 1

Question :-

solve for x,

$\sf{ \frac{x + 4}{ x - 4} - \frac{x - 4}{x + 4} = \frac{1 - {x}^{2} }{ {x}^{2} - 16 } }$

Answer :-

$\sf{x \: = \frac{ - 16 + \sqrt{148} }{2} \: or \: \frac{ - 16 - \sqrt{148} }{2} } \:$

Formula used :-

$\star \: \sf{ {(x + y)}^{2} = {x}^{2} + {y}^{2} + 2xy} \\ \\ \star \sf{ {( x - y)}^{2} = {x}^{2} + {y}^{2} - 2xy} \\ \\ \star \: \sf{ {x}^{2} - {y}^{2} = (x - y)(x + y)}$

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Solution :-

$\rightarrow \: \sf{ \frac{x + 4}{ x - 4} - \frac{x - 4}{x + 4} = \frac{1 - {x}^{2} }{ {x}^{2} - 16 } } \\ \: \\ \rightarrow \sf{ \frac{ {(x + 4)}^{2} - {(x - 4)}^{2} }{(x + 4)(x - 4)} = \frac{1 - {x}^{2} }{ {x}^{2} - 16 }} \\ \\ \rightarrow \sf{ \frac{ {x}^{2} + 16 + 8x - {x}^{2} - 16 + 8x }{ {x}^{2} - 16} = \frac{1 - {x}^{2} }{ {x}^{2} - 16 } }\: \\ \\ \rightarrow \sf{ 16x = 1 - {x}^{2} } \\ \\ \rightarrow \sf{ {x}^{2} + 16x - 1 = 0 }\\ \\ \rightarrow \sf{x \: = \frac{ - 16 + \sqrt{144 + 4} }{2} or \frac{ - 16 - \sqrt{144 + 4} }{2} } \\ \\ \rightarrow \sf{x \: = \frac{ - 16 + \sqrt{148} }{2} \: or \: \frac{ - 16 - \sqrt{148} }{2} }$

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