Question

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

Answer 1

Step-by-step explanation:

a=

2

3+

5

∴

a

1

=

3+

5

2

On rationaliging the denominator , we get ,

a

1

=

(3+

5

)(3−

5

)

2(3−

5

)

=

9−5

6−2

5

=

4

6−2

5

=

2

3−

5

Also,

(a+

a

1

)

2

=a

2

+

a

2

1

+2

Substituting the values of a



=

a

1

,

We get ,

`(

2

3+

5

+

2

3−

5

)

2

=(a

2

+

a

2

1

+2)

∴a

2

+

a

2

1

+2=(

2

3+

5

+3−

5

)

2

=(3)

2

=9

∴a

2

+

a

2

1

=9−2=7.

Answer 2

Step-by-step explanation:

$\bf \underline{Solution-}$

Given expression

$\bf \: a = \frac{5 + \sqrt{3} }{2}$

$\bf \therefore \: \frac{1}{a} = \frac{1}{ \frac{5 + \sqrt{3} }{2} } = \frac{2}{5 + \sqrt{3} }$

$\bf \: So, \: \frac{1}{a} = \frac{2}{5 + \sqrt{3} }$

The denomination = 5 + √3.

We know that

Rationalising factor of x + √y = x - √y.

So, rationalising factor of 5 + √3 = 5 - √3.

On both the denominator them

$\bf \: = \frac{2}{5 + \sqrt{3} } \times \frac{5 - \sqrt{3} }{5 - \sqrt{3} }$

$\bf = \frac{2(5 - \sqrt{3} )}{(5 + \sqrt{3} )(5 - \sqrt{3}) }$

• [(x + y)(x - y) = x² - y²
• Where, x = 5 and y = √3]

$\bf = \frac{2(5 - \sqrt{3}) }{( 5 {)}^{2} - ( \sqrt{3} {)}^{2} }$

$\bf = \frac{2(5 - \sqrt{3} )}{25 - 3}$

$\bf = \frac{2(5 - \sqrt{3} )}{22}$

$\bf = \frac{5 - \sqrt{3} }{2}$

Now, adding both values a and 1/a we get

$\bf \: a + \frac{1}{a} = \frac{5 + \sqrt{3} }{2} + \frac{5 - \sqrt{3} }{2}$

$\bf \: a + \frac{1}{a} = \frac{5 + \cancel{\sqrt{3}} + 5 - \cancel{ \sqrt{3}} }{2}$

$\bf \: a + \frac{1}{a} = \frac{ 5 + 5}{2}$

$\bf \: a + \frac{1}{a} = \cancel{\frac{10}{2} }$

$\bf \: a + \frac{1}{a} = 5$

Now, squaring on both sides, we get

$\bf \: \bigg( a + \frac{1}{a} \bigg) ^{2} = (5 {)}^{2}$

• (a+b)² = a² + 2ab + b².

$\bf \longrightarrow\: {a}^{2} + 2( \cancel{a}) \bigg( \frac{1}{ \cancel{a}} \bigg) + \bigg( \frac{1}{ a} \bigg) ^{2} = 25$

$\bf \longrightarrow \: {a}^{2} + 2 + \frac{1}{ {a}^{2} } = 25$

$\bf\longrightarrow \: {a}^{2} + \frac{1}{ {a}^{2} } = 25 - 2$

$\bf\longrightarrow \: {a}^{2} + \frac{1}{ {a}^{2} } = 23$

$\underline{\bf \: Hence \: the \: required \: value \: of \: {a}^{2} + \frac{1}{ {a}^{2} } is \: 23.}$