Question

If a=2+√5 and b= 1/a
then find a² + b²
Please ans

Answer 1

Answer:

a = 2 + \sqrt{5}a=2+

5

b = \frac{1}{a}b=

a

1

so,

b = \frac{1}{2 + \sqrt{5} }b=

2+

5

1

b = \frac{1 \times( 2 - \sqrt{5} )}{2 + \sqrt{5} \times (2 - \sqrt{5}) }b=

2+

5

×(2−

5

)

1×(2−

5

)

b = \frac{2 - \sqrt{5} }{ {2}^{2} - 5}b=

2

2

−5

2−

5

b = \frac{2 - \sqrt{5} }{4 - 5}b=

4−5

2−

5

b = \frac{2 - \sqrt{5} }{ - 1}b=

−1

2−

5

b = \sqrt{5 } - 2b=

5

−2

{a}^{2} + {b}^{2}a

2

+b

2

so put value of a and b in eq.

{(2 + \sqrt{5} )}^{2} + { (\sqrt{5 } - 2)}^{2}(2+

5

)

2

+(

5

−2)

2

we. get

{(a + b)}^{2} = {a}^{2} + {b}^{2} + 2b(a+b)

2

=a

2

+b

2

+2b

(4 + 5 + 2(4)( \sqrt{5} )\: \: + (4 + 5 +( - 2 \times 4 \times \sqrt{5} ))(4+5+2(4)(

5

)+(4+5+(−2×4×

5

))

(4 + 5 + 8 \sqrt{5} ) + (4 + 5 - 8 \sqrt{5} )(4+5+8

5

)+(4+5−8

5

)

9 + 99+9

18 ans