Question

$if \: x - \frac{1}{x} = 5 \: then \: x {}^{2} - \frac{1}{x {}^{2} } = |?| \:$
​

Answer 1

Answer:

5√29

Step-by-step explanation:

Given : x - 1/x = 5 ...(I)

Squaring both sides, we get

(x - 1/x)² = 5²

{ (a - b)² = a² + b² - 2ab

Here, a = x, b = 1/x }

(x)² + (1/x)² - 2(x)(1/x) = 25

x² + 1/x² - 2 = 25

x² + 1/x² = 25 + 2

x² + 1/x² = 27 ...(ii)

Now,

Squaring of (x + 1/x), we get

(x + 1/x)² = (x)² + (1/x)² + 2(x)(1/x)

{ (a + b)² = a² + b² + 2ab

Here, a = x, b = 1/x }

= x² + 1/x² + 2

= 27 + 2 [using (ii)]

= 29

x + 1/x = √29 ...(iii)

As we know that,

{ a² - b² = (a - b)(a + b)

Here, we take a = x, b = 1/x }

→ x² - 1/x² = (x - 1/x)(x + 1/x)

→ x² - 1/x² = (5)(√29) [using (i) and (iii)]

x² - 1/x² = 5√29