(i) Given equation is $x-(1)/(x)=3$
$implies(x^(2)-1)/(x)=3$
$impliesx^(2)-1=3ximpliesx^(2)-3x-1=0$
On comparing with $ax^(2)+bx+c=0$
a=1,b=-3 and c=-1
$becausex=(-b+-sqrt(b^(2)-4ac))/(2a)$
$impliesx=(-(3)+-sqrt((-3)^(2)-4xx1xx-1))/(2xx1)$
$impliesx=(3+-sqrt13)/(2)$
$becausex=(3+sqrt13)/(2) and (3-sqrt13)/(2)$
Hence, roots of the equation are $(3+sqrt13)/(2) and (3-sqrt13)/(2)$
(ii) Given equation is $(1)/(x+4)-(1)/(x-7)=(11)/(30)$
$implies((x-7)-(x+4))/((x+4)(x-7))=(11)/(30)implies(-11)/(x^(2)-7x+4x-28)=(11)/(30)$
$implies11(x^(2)-3x-28)=30xx(-11)$
$impliesx^(2)-3x-28=-30$
$impliesx^(2)-3x+2=0$
$impliesx^(2)-(2+1)x+2=0$
$impliesx^(2)-2x-x+2=0$
$impliesx(x-2)-1(x-2)=0$
$implies(x-2)(x-1)=0$
$impliesx=2 and x=1$
Hence, roots of the equation are 2 and 1.
