Question

If x =
$\sqrt{5} + 2$
, then
$(x + 1 \div x {}^{} ) {}^{2}$
Answer it correctly ✌​

Answer 1

Answer:

$(x + 1$ ÷ $x)^2$ $= 20$

Step-by-step explanation:

$(x + 1$ ÷ $x)^2$

$= (x + \frac{1}{x} )^2$

$= x^2 + 2.x.\frac{1}{x} + \frac{1}{x^2}$

$= (\sqrt{5} + 2) ^2 + 2 + \frac{1}{(\sqrt{5} + 2)^2}$

$= 5 + 2.\sqrt{5}.2 + 4 + 2 + \frac{1}{5 + 2.\sqrt{5}.2 + 4}$

$= 11 + 4\sqrt{5} + \frac{1}{9 + 4\sqrt{5}}$

$= 11 + 4\sqrt{5} + \frac{9 - 4\sqrt{5}}{(9 + 4\sqrt{5})(9 - 4\sqrt{5})}$

$= 11 + 4\sqrt{5} + \frac{9 - 4\sqrt{5}}{(9^2 - 4^2.5)}$  [using the identity $a^2 - b^2 = (a + b)(a - b)$]

$= 11 + 4\sqrt{5} + \frac{9 - 4\sqrt{5}}{(81 - 80)}$

$= 11 + 4\sqrt{5} + 9 - 4\sqrt{5}$

$= 11 + 9$

$= 20$

Answer 2

Given :

x = √5 + 2

To find :

Value of (x + 1/x)²

Solution :

x = √5 + 2

∴ 1/x = 1/(√5 + 2)

⇒ 1/x = 1(√5 - 2)/{(√5 + 2)(√5 - 2)}     [Rationalising the denominator]

⇒ 1/x = (√5 - 2)/{ (√5)² - 2²}

⇒ 1/x = (√5 - 2)/(5 - 4)

⇒ 1/x = (√5 - 2)/1

⇒ 1/x = √5 - 2

Now given expression :

⇒ (x + 1/x)² = [(√5 + 2) + (√5 - 2)]²

⇒ (x + 1/x)² = [√5 + 2 + √5 - 2]²

⇒ (x + 1/x)² = (2√5)²

⇒ (x + 1/x)² = 2² × (√5)²

⇒ (x + 1/x)² = 4 × 5

⇒ (x + 1/x)² = 20

∴ Required value of (x + 1/x)² = 20.