If x=2-√3, find the value of x^2+1/x^2.
Answers
Answer:
x=2-√3
1/x = 2+√3
x^2+1/x^2
(2-√3)^2 +( 2+√3)^2
4-4√3+3 +4+4√3+3
4+3+4+3
14
Answer:
Hence, the value of x^2 + 1/x^2 is 14.
Step-by-step explanation:
Given:-
x = 2 - √3
To find out:-
Value of x^2 + 1 / x^2
Solution:-
We have,
x = 2 - √3
∴ 1/x = 1/2-√3
The denomination = 2-√3
We know that
Rationalising factor of a-√b = a+√b
So, the rationalising factor of 2-√3 = 2+√3
On rationalising the denominator then
1/x = [1/(2-√3)]×[(2+√3)/2+√3)]
1/x = [1(2+√3)]/[(2+√3)(2+√3)]
1/x = (2+√3)/[(2-√3)(2+√3)]
Now, we will apply algebraic Identity in denominator because the denominator is in the form of
(a-b)(a+b) = a^2 - b^2
Where we have to put a = 2 and b = √3, we get
1/x = (2+√3)/[(2)^2 - (√3)^2]
1/x = (2+√3)/(4 - 3)
1/x = (2+√3)/1
1/x = 2+√3
Now, we have to add both values x and 1/x, we get
∴ x + 1/x = 2-√3+2+√3
√3 will be cancel out
x + 1/x = 2 + 2
x + 1/x = 4
Now, squaring on both, we get
(x +1/x)^2 = (4)^2
Using algebraic Identity,
(a+b)^2 = a^2 + 2ab + b^2
Where we have to put a = x and b = 1/x , we get
➟ x^2 + 2(x)(1/x) + (1/x)^2 = 16
➟ x^2 + 2 + (1/x)^2 = 16
➟ x^2 + 2 + 1/x^2 = 16
➟ x^2 + 1/x^2 = 16 - 2
➟ x^2 + 1/x^2 = 14
Answer:-
Hence, the value of x^2 + 1/x^2 is 14.
Used formulae:-
Rationalising factor of a-√b = a+√b
(a-b)(a+b) = a^2 - b^2
(a+b)^2 = a^2 + 2ab + b^2
- I hope it's help you. ☺️