Answer:
4
Step-by-step explanation:
Given
x + [tex]\frac{1}{x}[/tex] ← substitute x = 2 + [tex]\sqrt{3}[/tex]
= 2 + [tex]\sqrt{3}[/tex] + [tex]\frac{1}{2+\sqrt{3} }[/tex]
Rationalise the denominator of the fraction by multiplying the numerator/ denominator by the conjugate of the denominator.
The conjugate of 2 + [tex]\sqrt{3}[/tex] is 2 - [tex]\sqrt{3}[/tex] , thus
= 2 + [tex]\sqrt{3}[/tex] + [tex]\frac{2-\sqrt{3} }{(2+\sqrt{3} )(2-\sqrt{3}) }[/tex]
= 2 + [tex]\sqrt{3}[/tex] + [tex]\frac{2-\sqrt{3} }{4-3}[/tex]
= 2 + [tex]\sqrt{3}[/tex] + 2 - [tex]\sqrt{3}[/tex]
= 4
