Question

# If x = 2 + √3, find the value of x^{2} + frac{1}{x^{2}}

Solution

( begin{aligned} x=& text { (2) } 2+sqrt{3} x^{2}=(2+sqrt{3})^{2}=& 4+3+4 sqrt{3}=7+4 sqrt{3} text { Now, } frac{1}{x}=frac{1}{2+sqrt{3}} &=frac{1}{2+sqrt{3}} times frac{2-sqrt{3}}{2-sqrt{3}}=frac{2-sqrt{3}}{4-3}=2-sqrt{3} text { So } frac{1}{x^{2}}=(2-sqrt{3})^{2}=4+3-4 sqrt{3} &=7-4 sqrt{3} therefore x^{2}+frac{1}{x^{2}}=7+y / 3+7-4 sqrt{3} &=7+7 &=14 end{aligned} )