If x = 2+√3, find the value of x+1/x.
By BYJU'S Exam Prep
Updated on: September 25th, 2023
Rationalization is the name given to the procedure of rewriting a fraction in such a way that the denominator consists solely of rational numbers. To achieve this, the calculation of both x and 1/x should be done independently, followed by determining the final result.
Whenever the denominator of any term is irrational, it is necessary to rationalize the term. This involves multiplying and dividing the term by its conjugate, which refers to a number that, when multiplied by an irrational number, results in a rational number.
Table of content
Value of x+1/x
Solution
Given:
x = 2 + √3
To find the value of x + 1/x, we substitute the value of x into the expression.
x + 1/x = (2 + √3) + 1/(2 + √3)
To simplify the expression, we need to rationalize the denominator. We multiply both the numerator and denominator by the conjugate of the denominator, which in this case is (2 – √3).
1/x = (2 – √3) / (2 + √3)(2 – √3)
1/x = (2 – √3) / (2)2 – (√3)2
1/x = (2 – √3) / 4-3
1/x = (2 – √3) / 1
1/x = (2 – √3)
Therefore, x + 1/x = (2 + √3) + (2 – √3)
x + 1/x = 4.
Summary:
If x = 2+√3, find the value of x+1/x.
To determine the value of x + 1/x, we substitute the value of x into the expression and rationalize the denominator to simplify the expression. On solving we obtained, the value of x+1/x is 4.
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