Question

If [x=2+√5] find the value of x^2+1/x^2

Answer 1

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

$\green{\tt{\therefore{x^{2}+\frac{1}{x^{2}}=18}}}$

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

$\green{\underline \bold{Given :}} \\ \tt: \implies x = 2 + \sqrt{5} \\ \\ \red{\underline \bold{To \: Find : }} \\ \tt: \implies {x}^{2} + \frac{1}{ {x}^{2} } =?$

• According to given question :

$\bold{As \: we \: know \: that} \\ \tt: \implies {x}^{2} = ( 2 + \sqrt{5} )^{2} \\ \\ \tt: \implies {x}^{2} = {2}^{2} + ({ \sqrt{5} })^{2} + 4 \sqrt{5} \\ \\ \tt: \implies {x}^{2} = 4 + 5 + 4 \sqrt{5} \\ \\ \tt: \implies {x}^{2} =9 + 4 \sqrt{5} - - - - - (1) \\ \\ \bold{As \: we \: know \: that} \\ \tt: \implies \frac{1}{ {x}^{2} } = \frac{1}{(2 + \sqrt{5})^{2} } \\ \\ \tt: \implies \frac{1}{ {x}^{2} } = \frac{1 }{9 + 4 \sqrt{5} } \times \frac{9 - 4 \sqrt{5} }{9 - 4 \sqrt{5} } \\ \\ \tt: \implies \frac{1}{ {x}^{2} } = \frac{9 - 4 \sqrt{5} }{ {9}^{2} - {(4 \sqrt{5}) }^{2} } \\ \\ \tt: \implies \frac{1}{ {x}^{2} } = \frac{9 - 4 \sqrt{5} }{81 - 80} \\ \\ \tt: \implies \frac{1}{ {x}^{2} } =9 - 4 \sqrt{5}-----(2) \\ \\ \bold{For \: finding \: value : } \\ \tt: \implies {x}^{2} + \frac{1}{ {x}^{2} } =9 + 4 \sqrt{5} + 9 - 4 \sqrt{5} \\ \\ \tt: \implies {x }^{2} + \frac{1}{ {x}^{2} } =9 + 9 \\ \\ \green{\tt: \implies {x}^{2} + \frac{1}{ {x}^{2} } =18}$

Answer 2

GiveN:

• x = 2 + √5

To FinD:

• x² + 1/x²

SoLuTiOn:

→ x² + 1/x²

Substituting the value of x

→ (2 + √5)² + 1/(2 + √5)²

Using

(a + b)² = a² + 2ab + b²

→ 2² + 2(2)(√5) + (√5)² + 1/2² + 2(2)(√5) + (√5)²

→ 4 + 4√5 + 5 + 1/4 + 4√5 + 5

→ 9 + 4√5 + 1/9 + 4√5

→ (9 + 4√5)² + 1/9 + 4√5

Using the same formula

→ 9² + 2(9)(4√5) + (4√5)² + 1/9 + 4√5

→ 81 + 72√5 + 16(5) + 1/9 + 4√5

→ 162 + 72√5/9 + 4√5

Taking common

→ 18(9 + 4√5)/9 + 4√5

→ 18

Hence , x² + 1/x² = 18