Question

If x=2+√3, find the value of x²+1/x²​

Answer 1

ANSWER:

• Value of the above expression = 14

GIVEN:

• x = 2+√3

TO FIND:

• x²+1/x²

SOLUTION:

=> x = 2+√3.

Finding 1/x :

$\implies \: \dfrac{1}{x} = \dfrac{1}{2 + \sqrt{3} } \\ \\ \implies \: \dfrac{1}{x} = \dfrac{1}{2 + \sqrt{3} } \times \dfrac{2 - \sqrt{3} }{2 - \sqrt{3} } \\ \\ \implies \: \dfrac{1}{x} = \dfrac{2 - \sqrt{3} }{4 - 3} \\ \\ \implies \: \dfrac{1}{x} = 2 - \sqrt{3}$

Now finding :

= x²+1/x²

= (2+√3)²+(2-√3)²

= (2)²+(√3)²+2(2)(√3) + (2)²+(√3)²-2(2)(√3)

= 4+3+4√3 + 4+3-4√3

= 14

Value of the above expression = 14

NOTE:

Some important formulas:

(a+b)² = a²+b²+2ab

(a-b)² = a²+b²-2ab

(a+b)(a-b) = a²-b²

(a+b)³ = a³+b³+3ab(a+b)