Question

if X=(2+✓3), find the value of(x²+1/x²)²​

Answer 1

Answer:

196

Step-by-step explanation:

x = 2 + √3

∴ x² = ( 2 + √3)²

x² = (2)² + 2 (2) (√3) + (√3)²

x² = 4 + 3 + 4√3

x² = 7 + 4√3

Now,

1/x² = 1/ 7 + 4√3

..rationalizing

1/x² = $\frac{1}{(7 + 4\sqrt{3} )} * \frac{(7 - 4\sqrt{3} )}{(7 - 4\sqrt{3} )} = \frac{(7 - 4\sqrt{3} )}{(7)^2 - (4\sqrt{3})^2 } = \frac{(7 - 4\sqrt{3} )}{49 - 48} = (7 - 4\sqrt{3} )$

Finally,

(x²+1/x²)²​ = [(7 + 4√3) + (7 - 4√3)]²

               = [ 7 + 7]²

               = [ 14 ]²

               = 196

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Answer 2

Answer:

Step by step explanation:

Question:-

If x = 2 + √3, find the value of x² + 1/x²

To find:-

The value of x² + 1/x² = ?

Solution:-

Let's solve the problem

We have: x = 2+√3

∴ 1/x = 1/2+√3

The denominator is 2+√3. Multiplying the numerator and denomination by 2-√3, we get

➟ 1/2+√3 × 2-√3/2-√3

➟ 1(2-√3)/(2+√3)(2-√3)

⬤ Applying Algebraic Identity

(a+b)(a-b) = a² - b² to the denominator

We get,

➟ 2-√3 /(2)² - (√3)²

➟ 2 - √3 / 4 - 3

➟ 2 - √3 / 1

➟ 2 -√3

∴ x + 1/x = 2+√3 + 2-√3

x + 1/x = 2 + 2

x + 1/x = 4

Squaring on both sides we get,

(x + 1/x)² = (4)²

➟ x² + 2(x)(1/x) + (1/x)² = 16

➟ x² + 2 + 1/x² = 16

➟ x² + 1/x² = 16 - 2

➟ x² + 1/x² = 14

Answer:-

Hence, the value of x² + 1/x² = 14.

Used Formulae:-

(a+b)(a-b) = a² - b²

:)