Question

Find the values of a and b if

${ \frac{ \sqrt{7} - 1 }{ \sqrt{7} + 1} - \frac{ \sqrt{7} + 1}{\sqrt{7} - 1} = a + b \sqrt{7} }$
​

Answer 1

Answer:

$a = 0 \\ \\ b = \frac{ - 4}{6}$

Step-by-step explanation:

$\frac{ \sqrt{7} - 1}{ \sqrt{7} + 1 } - \frac{ \sqrt{7} + 1 }{ \sqrt{7} - 1} = a + b \sqrt{7} \\ \\ = \frac{ \sqrt{7} - 1}{ \sqrt{7} + 1} \times \frac{ \sqrt{7} - 1}{ \sqrt{7} - 1} \\ \\ = \frac{ (\sqrt{7} + 1) {}^{2} }{( \sqrt{7} - 1)( \sqrt{7} - 1) } \\ \\ = \frac{( \sqrt{7} ) {}^{2} + ( {1)}^{2} - 2( \sqrt{7} \times 1)}{( \sqrt{7} {)}^{2} - ( {1)}^{2} } \\ \\ = \frac{7 + 1 - 2 \sqrt{7} }{7 - 1} \\ \\ = \frac{8 - 2 \sqrt{7} }{6}$

$\frac{ \sqrt{7} + 1}{ \sqrt{7} - 1} \times \frac{ \sqrt{7} + 1}{ \sqrt{7} + 1 } \\ \\ = \frac{( \sqrt{7} + 1 {)}^{2} }{( \sqrt{7} + 1)( \sqrt{7} - 1) } \\ \\ = \frac{ (\sqrt{7} {)}^{2} + ( {1)}^{2} + 2( \sqrt{7}) }{7 - 1} \\ \\ = \frac{8 + 2 \sqrt{7} }{6}$

$\frac{8 - 2 \sqrt{7} }{6} - \frac{8 + 2 \sqrt{7} }{6} \\ \\ = \frac{8 - 2 \sqrt{7} - 8 - 2 \sqrt{7} }{6} \\ \\ = \frac{ - 4 \sqrt{7} }{6}$

$a = 0 \\ \\ b = \frac{ - 4}{6}$

Answer 2

Answer:

The value of a = 0 and b =- $\frac{2}{3}$

Step-by-step explanation:

Given that:

$\frac{ \sqrt{7} - 1 }{ \sqrt{7} + 1 } - \frac{ \sqrt{7} + 1}{ \sqrt{7} - 1} = a + b \sqrt{7}$

To find:

The value of a and b.

Solution:

$\frac{ \sqrt{7} - 1 }{ \sqrt{7} + 1 } - \frac{ \sqrt{7} + 1}{ \sqrt{7} - 1}$

Rationalising the denominator, we get

$\longrightarrow \: \frac{ \sqrt{7} - 1}{ \sqrt{7} + 1 } \times \frac{ \sqrt{7} - 1}{ \sqrt{7} - 1} - \frac{ \sqrt{7} + 1 }{ \sqrt{7} - 1 } \times \frac{ \sqrt{7} + 1 }{ \sqrt{7} + 1 }$

$\longrightarrow \: \frac{( \sqrt{7} - 1 {)}^{2} }{( \sqrt{7} + 1)( \sqrt{7} - 1 } - \frac{( \sqrt{7} + 1 {)}^{2} }{( \sqrt{7} - 1( \sqrt{7} + 1) }$

$\longrightarrow \: \frac{7 + 1 - 2 \sqrt{7} }{( \sqrt{7} {)}^{2} - {1}^{2} } - \frac{7 + 1 + 2 \sqrt{7} }{( \sqrt{7} {)}^{2} - {1}^{2} }$

$\longrightarrow \: \frac{8 - 2 \sqrt{7} }{7 - 1} - \frac{8 + 2 \sqrt{7} }{7 - 1}$

$\longrightarrow \: \frac{1}{6} (8 - 2 \sqrt{7} - 8 - 2 \sqrt{7} )$

$\longrightarrow \: 0 - \frac{4 \sqrt{7} }{6}$

$\longrightarrow \: 0 + ( \frac{ - 2}{3} ) \sqrt{7}$

$∴ \: 0 + (\frac{ - 2}{3} ) = a + b \sqrt{7}$

$\longrightarrow \: a = 0 \: \: and \: \: b = - \frac{2}{3}$

According to the question we find a = 0 and b = -$\frac{2}{3}$