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Answer 1

Question

$\Rightarrow \sf \dfrac{ \sqrt{x + 13} + \sqrt{x - 1} }{ \sqrt{x + 13} - \sqrt{x - 1} } = 7$

Answer= 19

Step by step explanation

$\Rightarrow \sf \dfrac{ \sqrt{x + 13} + \sqrt{x - 1} }{ \sqrt{x + 13} - \sqrt{x - 1} } = 7$

${\Rightarrow\sf \dfrac{ (\sqrt{x + 13} + \sqrt{x - 1} )(\sqrt{x + 13} + \sqrt{x - 1} )}{ (\sqrt{x + 13} - \sqrt{x - 1} )(\sqrt{x + 13} + \sqrt{x - 1} )} = 7 }$

${\Rightarrow\sf \dfrac{(\sqrt{x + 13} + \sqrt{x - 1} )^{1 + 1} }{ \sqrt{x + 13} \times \sqrt{x + 13} + \sqrt{x + 13} \times \sqrt{x - 1} - \sqrt{x - 1} \times \sqrt{x + 13} - \sqrt{x - 1} \times \sqrt{x - 1} } = 7 }$

${\Rightarrow\sf \dfrac{(\sqrt{x + 13} + \sqrt{x - 1} )^{2} }{ \sqrt{(x + 13) (x + 13) } + \sqrt{(x + 13) {(x - 1)}} - \sqrt{(x - 1) (x + 13)} - \sqrt{(x - 1) (x - 1) } } = 7 }$

${\Rightarrow\sf \dfrac{(\sqrt{x + 13} + \sqrt{x - 1} )^{2} }{ \sqrt{(x + 13) ^{2} } + \sqrt{(x + 13) {(x - 1)}} - \sqrt{(x - 1) (x + 13)} - \sqrt{(x - 1) ^{2} } } = 7 }$

${\Rightarrow\sf \dfrac{(\sqrt{x + 13} + \sqrt{x - 1} )^{2} }{ {(x + 13)} + \sqrt{(x + 13) {(x - 1)}} - \sqrt{(x - 1) (x + 13)} - (x - 1) } = 7 }$

${\Rightarrow\sf \dfrac{(\sqrt{x + 13} + \sqrt{x - 1} )^{2} }{ {(x + 13) ^{ \frac{2}{2} } } + \sqrt{(x + 13) {(x - 1)}} - \sqrt{(x - 1) (x + 13)} - (x - 1) ^{ \frac{2}{2} } } = 7 }$

${\Rightarrow\sf \dfrac{(\sqrt{x + 13} + \sqrt{x - 1} )^{2} }{ {(x + 13)} + \sqrt{(x + 13) {(x - 1)}} - \sqrt{(x + 13) (x - 1)} - (x - 1) } = 7 }$

${\Rightarrow\sf \dfrac{(\sqrt{x + 13} + \sqrt{x - 1} )^{2} }{ {(x + 13)} \cancel{+ \sqrt{(x + 13) {(x - 1)}}} \cancel {- \sqrt{(x - 1) (x + 13)} } - (x - 1) } = 7 }$

${\Rightarrow\sf \dfrac{(\sqrt{x + 13} + \sqrt{x - 1} )^{2} }{ {(x + 13)} {- (x - 1) }}} = 7$

${\Rightarrow\sf \dfrac{(\sqrt{x + 13} + \sqrt{x - 1} )^{2} }{ {x + 13} { - x + 1 }}} = 7$

${\Rightarrow\sf \dfrac{(\sqrt{x + 13} + \sqrt{x - 1} )^{2} }{14}} = 7$

${\Rightarrow\sf (\sqrt{x + 13} + \sqrt{x - 1} )^{2} } =98$

${\Rightarrow\sf \{(\sqrt{x + 13} + \sqrt{x - 1} )^{2} \} ^{ \frac{1}{2} } =98 ^{ \frac{1}{2} } }$

${\Rightarrow\sf (\sqrt{x + 13} + \sqrt{x - 1} ) \ ^{ 2 \times \frac{1}{2} } =(2 \times 7^{2} )^{ \frac{1}{2} } }$

${\Rightarrow\sf \sqrt{x + 13} + \sqrt{x - 1} ) =2 ^{ \frac{1}{2} } \times 7 }$

${\Rightarrow\sf \sqrt{x + 13} + \sqrt{x - 1} =7 \times 2 ^{ \frac{1}{2} } }$

${\Rightarrow\sf (\sqrt{x + 13} + \sqrt{x - 1} ) + ( - \sqrt{x - 1}) =7 \times 2 ^{ \frac{1}{2} } + ( - \sqrt{x - 1} ) }$

${\Rightarrow\sf (\sqrt{x + 13}) ^{2} =(7 \times 2 ^{ \frac{1}{2} } - \sqrt{x - 1} ) ^{2} }$

${\Rightarrow\sf x + 13 =(7 \times 2 ^{ \frac{1}{2} } - \sqrt{x - 1} ) ^{2} }$

${\Rightarrow}\sf x + 13 =(7 \times 2 ^{ \frac{1}{2} } )^{2} + 2 \times 7 \times 2 ^{ \frac{1}{2} } ( - \sqrt{x - 1} ) + ( - \sqrt{x - 1} ) ^{2}$

${\Rightarrow}\sf x + 13 =49 \times 2 ^{ \frac{2}{2} } - 14 \times 2 ^{ \frac{1}{2} } \times \sqrt{x - 1} + (x - 1) { }^{ \frac{2}{2} }$

${\Rightarrow}\sf x + 13 =98 - 14 \times 2 ^{ \frac{1}{2} } \times \sqrt{x - 1} + x - 1$

${\Rightarrow}\sf x + 13 = - 14 \times 2 ^{ \frac{1}{2} } \times \sqrt{x - 1} + x + 98 - 1$

${\Rightarrow}\sf \cancel{ x} + 13 = - 14 \times 2 ^{ \frac{1}{2} } \times \sqrt{x - 1} + \cancel{ x} + 97$

${\Rightarrow}\sf 13 -97 = - 14 \times 2 ^{ \frac{1}{2} } \times \sqrt{x - 1} + 97 - 97$

${\Rightarrow}\sf - 84 = - 14 \times 2 ^{ \frac{1}{2} } \times \sqrt{x - 1}$

${\Rightarrow}\sf (- 84) {}^{2} =( - 14 \times 2 ^{ \frac{1}{2} } \times \sqrt{x - 1} ) {}^{2}$

${\Rightarrow}\sf (- 84)^{2} = 14 ^{2} \times (2 ^{ \frac{1}{2} } ) ^{2} \times (\sqrt{x - 1} ) {}^{2}$

${\Rightarrow}\sf (- 84)^{2} = 14 ^{2} \times 2( x - 1)$

${\Rightarrow}\sf (- 84)^{2} = (14 ^{2} \times 2)( x - 1)$

${\Rightarrow}\sf (- 84)^{2} = 14 ^{2} \times 2x + 14^{2} \times 2( - 1)$

${\Rightarrow}\sf 7056 = 392x - 392$

${\Rightarrow}\sf 7056 + 392 = 392x$

${\Rightarrow}\sf 7448 = 392x$

${\Rightarrow}\sf x = \frac{7448}{392}$

$\gray{\sf ~~~ x = 19}$

Solution check follows.

• If you wish to see whether any invalid solutions have been removed, please scroll to the bottom of the page

• $\Rightarrow \sf \dfrac{ \sqrt{x + 13} + \sqrt{x - 1} }{ \sqrt{x + 13} - \sqrt{x - 1} } = 7$

• $\Rightarrow \sf \dfrac{ \sqrt{19+13} + \sqrt{19- 1} }{ \sqrt{19 + 13} - \sqrt{19 - 1} } = 7$

• $\Rightarrow \sf \dfrac{ \sqrt{32} + \sqrt{18} }{ \sqrt{32} - \sqrt{18} } = 7$

• $\Rightarrow \sf \dfrac{ 4√2 + 3√2}{ 4√2 - 3√2 } = 7$

• $\Rightarrow \sf \dfrac{ 4 + 3}{ 4- 3 } = 7$

• $\Rightarrow \sf 7= 7$

• $\Rightarrow \sf 0= 0$

X=19

Answer 2

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