Question

if a=√5+3/√5-3 and b=√5-3/√5+3 , then find the value of a
${a +}^{2} + {b}^{2} + ab$
please give the full explanation of the answer. it's a humble request to all....
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Answer 1

Answer:

50

Explanation:

Given,

$\bullet \: a = \dfrac{ \sqrt{5} + 3}{ \sqrt{5} - 3}$

And,

$\bullet \: b = \dfrac{ \sqrt{5} - 3}{ \sqrt{5} + 3}$

We need to find the value of${a}^{2} + {b}^{2} + ab$

Rationalising ‘a’ as follows :-

$\implies \: a = \dfrac{ \sqrt{5} + 3 }{ \sqrt{5} - 3 } \times \dfrac{ \sqrt{5} + 3 }{ \sqrt{5} + 3 }$

$= \dfrac{( \sqrt{5} + 3)( \sqrt{5} + 3) }{( \sqrt{5} - 3)( \sqrt{5} - 3) }$

$= \dfrac{5 + 3 \sqrt{5} + 3 \sqrt{5} + 9 }{ {( \sqrt{5}) }^{2} + {(3)}^{2} }$

$= \dfrac{14 + 6\sqrt{5} }{5 - 9}$

$= \dfrac{14 + 6\sqrt{5} }{ - 4}$

$= \dfrac{14 - 6 \sqrt{5} }{4} \bf . \: . \: . \: . \: .(i)$

And also, rationalising ‘b’

$\implies \: b = \dfrac{ \sqrt{5} - 3 }{ \sqrt{5} + 3 } \:$

$= \dfrac{ (\sqrt{5} - 3)}{( \sqrt{5} + 3) } \times \dfrac{( \sqrt{5} - 3) }{( \sqrt{5} - 3)}$

$= \dfrac{( \sqrt{5} - 3)( \sqrt{5} - 3) }{( \sqrt{5} + 3)( \sqrt{5} - 3) }$

$= \dfrac{5 - 3 \sqrt{5} - 3 \sqrt{5} + 9}{ {( \sqrt{5} )}^{2} - {(3)}^{2} }$

$= \dfrac{14 - 6 \sqrt{5} }{5 - 9}$

$= \dfrac{14 - 6 \sqrt{5} }{ - 4}$

$= \dfrac{14 + 6 \sqrt{5} }{4} \bf. \: . \: . \: . \: . \: (ii)$

Now,

$\longrightarrow \: {a}^{2} + {b}^{2} + ab$

By the identity : $a^2 + b^2 = (a + b)^2 - 2ab$

$\longrightarrow \: ({a} + {b)}^{2} - 2ab+ ab$

From (i) and (ii) :-

${\longrightarrow \: \bigg( \dfrac{14 - 6 \sqrt{5}}{4} + \dfrac{14 + 6 \sqrt{5} }{4} \bigg)^{2} + \bigg( \dfrac{14 - 6 \sqrt{5} }{4}\bigg) \bigg( \dfrac{14 + 6 \sqrt{5} }{4} \bigg)}$

${\longrightarrow \: \bigg( \dfrac{14 - 6 \sqrt{5} + 14 + 6 \sqrt{5} }{4}\bigg)^{2} + \bigg( \dfrac{(14 - 6 \sqrt{5} )(14 + 6 \sqrt{5} )}{4 \times 4} \bigg)}$

${\longrightarrow \: \bigg( \dfrac{28}{4}\bigg)^{2} + \bigg( \dfrac{(14)^{2} - ( 6 \sqrt{5} )^{2} }{16} \bigg)}$

${\longrightarrow \: \bigg( 7\bigg)^{2} + \bigg( \dfrac{196 - 180}{16} \bigg)}$

${\longrightarrow \: 49 + \bigg( \dfrac{16}{16} \bigg)}$

${\longrightarrow \: 49 + 1}$

$\longrightarrow \: 50$

Required answer: 50

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