Math, asked by martinthegamer360, 9 days ago

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

Answers

Answered by BrainlyConqueror0901
17

\blue{\bold{\underline{\underline{Answer:}}}}

\green{\tt{\therefore{a^{2}+\frac{1}{a^{2}}=\frac{40-9\sqrt{7}}{2}}}}\\

\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}

 \green{\underline \bold{Given :}} \\  \tt:  \implies a = \frac{3 +  \sqrt{7} }{ 2}  \\  \\ \red{\underline \bold{To \: Find:}} \\  \tt:  \implies  {a}^{2}  +  \frac{1}{ {a}^{2} }  =?

• According to given question :

 \bold{As \: we \: know \: that} \\  \tt:  \implies  {a}^{2}  =  ( \frac{3 +  \sqrt{7} }{2})^{2}  \\  \\ \tt:  \implies  {a}^{2}  =   \frac{ {3}^{2} +  {( \sqrt{7}) }^{2}  + 2 \times 3 \times  \sqrt{7}  }{4}  \\  \\ \tt:  \implies  {a}^{2}  =   \frac{9 + 7 + 6 \sqrt{7} }{4}  \\  \\ \tt:  \implies  {a}^{2}  =   \frac{16 + 6 \sqrt{7} }{4}  \\  \\ \tt:  \implies  {a}^{2}  =  \frac{8 + 3 \sqrt{7} }{2}  -  -  -  -  -  (1) \\  \\  \bold{As \: we \: know \: that} \\ \tt:  \implies   \frac{1}{ {a}^{2} }   =  \frac{1}{( { \frac{3 +  \sqrt{7} }{2}) }^{2} }  \\  \\ \tt:  \implies   \frac{1}{ {a}^{2} }   =  \frac{4}{9 + 7 + 6 \sqrt{7} }  \\  \\ \tt:  \implies   \frac{1}{ {a}^{2} }   =   \frac{2}{8 + 3 \sqrt{7} }  \\  \\ \tt:  \implies   \frac{1}{ {a}^{2} }   =   \frac{2}{8 + 3 \sqrt{7} }  \times  \frac{8 -  3\sqrt{7} }{8 - 3 \sqrt{7} }  \\  \\ \tt:  \implies   \frac{1}{ {a}^{2} }   =  \frac{16 - 6 \sqrt{7} }{64 - 63}  \\  \\ \tt:  \implies   \frac{1}{ {a}^{2} }   = 16 - 6 \sqrt{7}  -  -  -  -  - (2) \\  \\  \bold{For \: finding \: value } \\ \tt:  \implies   {a}^{2}   + \frac{1}{ {a}^{2} }    =  \frac{8 + 3 \sqrt{7} }{2}  + 16 - 6 \sqrt{7}  \\  \\ \tt:  \implies   {a}^{2}   + \frac{1}{ {a}^{2} }    =  \frac{8 + 3 \sqrt{7} + 32 - 12 \sqrt{7}  }{2}  \\  \\  \green{\tt:  \implies   {a}^{2}   + \frac{1}{ {a}^{2} }    =  \frac{40 - 9 \sqrt{7} }{2} }

Answered by ItzArchimedes
31

GIVEN:

  • a = (3 + √7)/2

TO FIND:

  • a² + ( 1/a² )

SOLUTION:

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

Butterfly method: 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 , + 1/ = 40 - 97/2

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