# If a = (3+√7)/2 , then find the value of a² + 1/a²

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__GIVEN____:__

- a = (3 + √7)/2

__TO____ ____FIND____:__

- a² + ( 1/a² )

__S____O____L____U____T____I____O____N____:__

Substituting the value of a

→ [( 3 + √7 )/2]² + { 1/[( 3 + √7 )/2]²

→ ( 3 + √7 )²/4 + { 1/( 3 + √7 )²/4

Simplifying using

(a + b)² = a² + 2ab + b²

→ [3² + 2(3)(√7) + (√7)²/4] + [ 4/3² + 2(3)(√7) + (√7)²]

→ (9 + 6√7 + 7/4) + (4/9 + 6√7 + 7)

→ ( 16 + 6√7/4 ) + ( 4/16 + 6√7 )

Simplifying using **Butterfly**** ****method**

**Butterf****ly**** ****meth****od****:**** **a/b + c/d = (ad + bc)/bd

→ (16 + 6√7)² + 4²/4(16 + 6√7)

Simplifying using

(a + b)² = a² + 2ab + b²

→ 16² + 2(16)(6√7) + (6√7)² + 4²/64 + 24√7

→ 524 + 19√7/64 + 24√7

→ 4(131 + 48√7)/8(8 + 3√7)

→ 131 + 48√7/2(8 + 3√7)

Rationalizing the denominator

→ [131 + 48√7/2(8 + 3√7)] × [8 - 3√7/8 - 3√7 ]

→ 131 + 48√7(8 - 3√7)/2[8² + (3√7)²]

Simplifying

→ 1048 - 393√7 + 384√7 - 1008/2(64 - 63)

→ 40 - 9√7/2(1)

→ 40 - 9√7/2

**Hence**** ****,**** ****a²**** ****+**** ****1****/****a²**** ****=**** ****4****0**** ****-**** ****9****√****7****/****2**

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