Question

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

Answer 1

Answer:

2

Step-by-step explanation:

x=2+$\sqrt{3}$

Therefore,

$x^{2} +1/x^{2}$  

=(x+1/x)^2 - 2*x*1/x

=(x+1/x)^2 - 2 {since x*1/x = 1}

putting the values,

=(2+$\sqrt{3}$ + 1/2+$\sqrt{3}$)^2 - 2

=[(2+$\sqrt{3}$)^2 + 1]/2+$\sqrt{3}$ - 2

=[(4+2*2*$\sqrt{3}$+3)+1]/2+$\sqrt{3}$ - 2

=[(4+4$\sqrt{3}$+3+1]/2+$\sqrt{3}$ - 2

=[(8+4$\sqrt{3}$)/2+$\sqrt{3}$] - 2

=[4(2+$\sqrt{3}$)/2+$\sqrt{3}$]-2

=4(1)-2

=4-2

=2

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

:)