Question

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

Answer 1

Step-by-step explanation:

This is the long way to find the value of x² + 1/x², substitute the given x = 2 + √3 into the variable x.

x² + 1/x²

= (2 + √3)² + 1/(2 + √3)²

= [2² + 2(2)(√3) + (√3)²] + 1/[2² + 2(2)(√3) + (√3)²]

= (4 + 4√3 + 3) + 1/(4 + 4√3 + 3)

= (7 + 4√3) + 1/(7 + 4√3)

= (7 + 4√3)²/(7 + 4√3) + 1/(7 + 4√3)

= ((7 + 4√3)² + 1)/(7 + 4√3)

= (7² + 2(7)(4√3) + (4√3)² + 1)/(7 + 4√3)

= (49 + 56√3 + 48 + 1)/(7 + 4√3)

= (98 + 56√3)/(7 + 4√3)

= 14(7 + 4√3)/(7 + 4√3)

= 14

Answer 2

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

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