Question

$\frac{x}{4} \sqrt{144 - {x -}^{2} } - \frac{x}{4} \sqrt{324 - {x }^{2} } = 36 \sqrt{2}$
Solve for x.​

Answer 1

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Refer from the attachment provided.

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Answer 2

$\frac{x}{4} \sqrt{144 - {x }^{2} } - \frac{x}{4} \sqrt{324 - {x }^{2} } = 36 \sqrt{2}$

$= > \frac{x}{4} ( \sqrt{144 - {x}^{2} } - \sqrt{324 - {x}^{2} } ) = 36 \sqrt{2}$

$= > \sqrt{ 144 - {x}^{2} } - \sqrt{ 324 - {x}^{2} } = 36 \sqrt{2} \times \frac{4}{x}$

$= > \sqrt{ 144- {x}^{2} } - \sqrt{ 324- {x}^{2} } = \frac{144 \sqrt{2} }{x}$

$= > \sqrt{ 144- {x}^{2} } = \sqrt{ 324 - {x}^{2} } + \frac{144 \sqrt{2} }{x}$

squaring both sides,

$= >( { \sqrt{ 144 - {x}^{2} } } \: )^{2} = ({ \sqrt{ 324 - {x}^{2} } + \frac{144 \sqrt{2} }{x} })^{2}$

$= > 144- {x}^{2} = 324 - {x}^{2} + \frac{ ({144 \sqrt{2} })^{2} }{ {x}^{2} } - 2 \times \sqrt{ 324- {x}^{2} } \times \frac{144 \sqrt{2} }{x}$

$= > 144 - 324 = \frac{( {144 \sqrt{2} )}^{2} }{ {x}^{2} } - 2 \sqrt{324 - {x}^{2} } \frac{144 \sqrt{2} }{x}$

$= > - 180 = \frac{144 \sqrt{2} }{x}( \frac{144 \sqrt{2} }{x} - 2 \sqrt{324 - {x}^{2} } )$

$= > - 180 \times \frac{x}{144 \sqrt{2} } = \frac{144 \sqrt{2} }{x} - 2 \sqrt{324 - {x}^{2} }$

$= > - \frac{5x}{4 \sqrt{2} } = \frac{144 \sqrt{2} }{x} - 2 \sqrt{324 - {x}^{2} }$

$= > 2 \sqrt{324 - {x}^{2} } = \frac{144 \sqrt{2} }{x} + \frac{5x}{4 \sqrt{2} }$

Squaring both sides again ,

$= > ( {2 \sqrt{324 - {x}^{2} }} \: )^{2} =({\frac{144 \sqrt{2} }{x} + \frac{5x}{4 \sqrt{2} }})^{2}$

$= > 4(324 - {x}^{2} ) = \frac{ {(144 \sqrt{2} })^{2} }{ {x}^{2} } + \frac{( {5x)}^{2} }{ ({4 \sqrt{2} })^{2} } + 2 \times \frac{144 \sqrt{2} }{x} \times \frac{5x}{4 \sqrt{2} }$

$= > 4 \times 324 - 4 {x}^{2} = \frac{ ({144 \sqrt{2} })^{2} }{ {x}^{2} } + \frac{25 {x}^{2} }{32} + 72 \times 5$

$= > 1296 = 4 {x}^{2} + \frac{25 {x}^{2} }{32} + \frac{20736 \sqrt{2} }{ {x}^{2} } + 360$

$= > 4 {x}^{2} + \frac{25 {x}^{2} }{32} = \frac{20736 \sqrt{2} }{ {x}^{2} } - 936$

$= > \frac{128 {x}^{2} + 25 {x}^{2} }{32} = \frac{20736 \sqrt{2} - 936 {x}^{2} }{ {x}^{2} }$

$= > {x}^{2} ( 153 {x}^{2} ) = 32(20736 \sqrt{2} - 936 {x}^{2} )$

=> 17×9x⁴ = 32×9(2304√2-104x²)

→ 17x⁴ = 73728√2-3328x²

→ 17x⁴+3328x²-73728 = 0