Question

if a= 3+√5/2, then find the value of a²+1/a²​

Answer 1

Answer:

$\bf\ \: answer \implies \: 7$

✏Question✒a= 3+√5/2, then find the value of a²+1/a²

✏to find✒a²+1/a²

✏solution✒

$\bf\ a = \frac{3 + \sqrt{5} }{2} \\ \\ \bf\ \implies: \frac{1}{a} = \frac{2}{3 + \sqrt{5} } \\ \\ \bf\ \implies: \frac{1}{a} = \frac{2(3 - \sqrt{5)} }{(3 + \sqrt{5} )(3 - \sqrt{5} )} \\ \\ \bf\ \implies: \frac{1}{a} = \frac{6 - 2 \sqrt{5} }{ ({3})^{2} - {( \sqrt{5} )}^{2} } \\ \\ \bf\ \implies: \frac{1}{a} = \frac{6 - 2 \sqrt{5} }{9 - 5} \\ \\ \bf\ \implies: \frac{1}{a} = \frac{2(3 - \sqrt{5)} }{4} \\ \\ \bf\boxed{ \implies: \frac{1}{a} = \frac{3 - \sqrt{5} }{2} } \\ \\ \huge\bf{ now \leadsto: } \\ \\ \bf\ \implies: {a}^{2} + \frac{1}{ {a}^{2} } \\ \\ \bf\ \implies: { (\frac{3 + \sqrt{5} }{2}) }^{2} + { (\frac{3 - \sqrt{5} }{2} )}^{2} \\ \\ \bf\ \implies: \frac{ {(3 - \sqrt{5} )^{2} + (3 + \sqrt{5)} }^{2} }{ {2}^{2} } \\ \\ \bf\ \implies: \frac{9 - \cancel{6 \sqrt{5} }+ 5 + 9 + \cancel{6 \sqrt{5}} + 5 }{4} \\ \\ \bf\ \implies: \frac{28}{4} \\ \\ \bf\boxed{ \implies:7 \: \: \bigstar }$

Answer 2

Answer: 7

Step by step explanation:

Given: a = 3 + √5 / 2

Therefore,

1 / a = 2 / 3 + √5

= 2 × ( 3 - √5 ) / ( 3 + √5 ) × ( 3 - √5 )

= 2 × ( 3 - √5 ) / ( 3 )² - ( √5 )²

= 3 - √5 / 2a + 1 /a

= 3 + √5 /2 + 3 - √5 /2

= 6 / 2

a + 1/a = 3

Using identity: (a + b)² = a² + 2ab + b²

(a + 1 /a)² = a² + 1/a² + 2 (a) × (1/a)

=> (a + 1 /a)² - 2 (a) × (1/a) = a² + 1/a²

=> (3)² - 2 = a² + 1/a

=> (3)² - 2 = a² + 1/a²

=> (9) - 2 = a² + 1/a²

7 = a² + 1 /a²

Thus, answer to your question is 7.