# Number system practice

Operation on real numbers & algebraic identities

#### Explanatory Answer : Method 2

In this method let us find x2 and its reciprocal and add it and check whether we get the same answer

x2 = ${$$frac{3 - {$sqrt{13}}}{2}$^2}$\$ = ${$$frac{9 + 13 - 6{\sqrt{13}}}{4}$}$\$ = ${$$frac{22 - 6{\sqrt{13}}}{4}$}$\$ = ${$$frac{11 - 3{\sqrt{13}}}{2}$}$\$ $\frac{1}{x^2}\\$ = ${$$frac{2}{11 - 3{\sqrt{13}}}$}$\$ $x^2 + \frac{1}{x^2}\\$ = ${$$frac{11 - 3{\sqrt{13}}}{2}$}$\$ + ${$$frac{2}{11 - 3{\sqrt{13}}}$}$\$ = $\frac{$11 - 3{$sqrt{13}}$^2 + 4}{2${11 - 3{$sqrt{13}}})}$\$ = $\frac{$121 + 117 - 66{$sqrt{13}}$ + 4}{2${11 - 3{$sqrt{13}}})}$\$ = $\frac{$242 - 66{$sqrt{13}}$}{2${11 - 3{$sqrt{13}}})}$\$ Take 22 as a term common in the numerator and rewrite the expression = $\frac{22$11 - 3{$sqrt{13}}$}{2${11 - 3{$sqrt{13}}})}$\$ Cancel ${$11 - 3{\sqrt{13}}$} $\$ in both the numerator and denominator
$x^2 + \frac{1}{x^2}\\$ = $\frac{22}{2} \\$ = 11.