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

By K Balaji|Updated : November 12th, 2022

Given

x = 2 + √3 …. (1)

1/x = 1/ 2 + √3

Let's undertake a denominator rationalisation.

Since, the 2 - √3 is Rationalizing factor of 2 + √3, multiply both numerator and the denominatoe with 2 - √3

= [1 x (2 - √3)]/ (2 + √3) (2 - √3)

= (2 - √3)/ (2)2 - (√3)2

= (2 - √3)/ (4 - 3)

= (2 - √3)/1

1/x = 2 - √3 …. (2)

Now,

x + 1/x = 2 + √3 + 2 - √3 = 4

Rationalisation

A radical or imaginary number can be removed from the denominator of an algebraic fraction by a procedure known as rationalization. That is, eliminate the radicals from a fraction to leave only a rational integer in the denominator.

A radical is an expression that makes use of a root, like the square or cube root.

We are tracing the origin of the word "radicand."

Mathematical conjugate: A second exact binomial with the opposite sign between its two terms is referred to as a conjugate of any binomial.

How to rationalize a denominator?

In order to eliminate the radicals in the denominator, multiply the denominator and numerator by an appropriate radical in step one.

Step 2: Verify that the fraction's surds are all expressed in their simplest form.

Step 3: If additional fractional simplification is required, do so.

Summary:-

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

If x = 2 + √3, the value of x + 1/x is 4

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