Question

step by step answer.....

Answer 1

Good Morning!

( x² - 1 )/x = 1/2 { Given }

x²/x - 1/x = 1/2

x - 1/x = 1/2

SQUARING ON BOTH SIDE'S WE HAVE

( x - 1/x )² = ( 1/2 )²

x² + 1/x² - 2x ( 1/x ) = 1/4

Becoz ( a - b )² = a² + b² - 2ab

x² + 1/x² = (1/4) + 2

x² + 1/x² = 9/4

MULTIPLY BOTH SIDES BY 4 WE HAVE

4( x² + 1/x² ) = (9/4) × 4

4x² + 4/x² = 9

SO,

4x² + 4/x² = 9

Answer 2

Given numeric value of ( x^2 - 1 ) / x = 1 / 2.

$\implies \dfrac{x {}^{2} - 1}{x} = \dfrac{1}{2} \\ \\ \implies \dfrac{ {x}^{2} }{x} - \dfrac{1}{x} = \dfrac{1}{2} \\ \\ \implies x - \dfrac{1}{x} = \dfrac{1}{2} \\ \\ \implies \bigg( x - \dfrac{1}{x} \bigg) \times 2 = \dfrac{1}{2} \times 2 \\ \\ \implies 2 x - \dfrac{2}{x} = 1$

Square on both sides : -

$\implies \bigg \{2 x - \dfrac{2}{x} \bigg \} {}^{2} = {1}^{2}$

From the properties of indices :

• ( a - b )^2 = a^2 + b^2 + 2ab

$\implies (2 x) {}^{2} + \bigg( \dfrac{2}{x} \bigg) {}^{2} - 2 \bigg(2x \times \dfrac{2}{x } \bigg) = {1}^{2} \\ \\ \implies 4 {x}^{2} + \frac{4}{ {x}^{2} } - 2(4) = 1 \\ \\ \implies 4 {x}^{2} + \frac{4}{ {x}^{2} } = 1 + 8 \\ \\ \implies 4 {x}^{2} + \frac{4}{ {x}^{2} } = 9$

Hence the required numeric value of $4 {x}^{2} + \frac{4}{ {x}^{2} }$ is 9.