Question

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

Answer 1

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

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

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

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

• According to given question :

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

Answer 2

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 , a² + 1/a² = 40 - 9√7/2