Question

If a and b are rational number, find a and b. (b)

Answer 1

Answer:

the value of a=18

b=0

this is your answer

Answer 2

We have,

$\frac{ \sqrt{5} - 2 }{ \sqrt{5} + 2} - \frac{ \sqrt{5} + 2}{ \sqrt{5} - 2 } = a + b \sqrt{5} \\ \\ \texttt{ Rationalising the denominator} \\ \\\scriptsize{\frac{ \sqrt{5} - 2 \times ( \sqrt{5} - 2) }{ \sqrt{5} + 2 \: \times ( \sqrt{5} - 2) } - \frac{ \sqrt{5} + 2 \times ( \sqrt{5} + 2) }{ \sqrt{5} - 2 \times ( \sqrt{5} + 2)}} \\ \\ = \small \frac{( { \sqrt{5} - 2) }^{2} }{ {( \sqrt{5} )}^{2} - (2) {}^{2} } - \frac{ {( \sqrt{5} + 2) }^{2} }{ {( \sqrt{5}) }^{2} - {(2)}^{2} } \\ \\ = \scriptsize\frac{ (\sqrt{5}) {}^{2} + ({2})^{2} - 2(\sqrt{5} )(2)}{5 - 4} - \frac{ {( \sqrt{5} )}^{2} + (2) {}^{2} + 2( \sqrt{5})(2) }{5 - 4} \\ \\ \texttt{Identities used:}\\(a+b)^2=a^2+b^2+2ab \\(a-b)^2= a^2+b^2-2ab\\(a+b)(a-b)=a^2-b^2</p><p>\\ \\ = \frac{5 + 4 - 4 \sqrt{5} }{1} - \frac{5 + 4 + 4 \sqrt{5} }{1} \\ \\ = 9 - {4 \sqrt{5}} -( 9 + {4 \sqrt{5} } ) \\ \\ = \cancel{9} - 4 \sqrt{5} - \cancel{ 9} - 4 \sqrt{5} \\ \\ \implies - 8 \sqrt{5} = a + b \sqrt{5}$

On comparing it with a +b√5

We get,

$\boxed{ \green{a = 0}} \\ \texttt{and} \\ \boxed{ \green{b = - 8}}$