Find the roots of the following equations : (i) `x-1/x=3,x!=0` (ii) `1/(x+4)-1/(x-7)=(11)/(30),x!=-4,7`

Find the roots of the following equations : (i) `x-1/x=3,x!=0` (ii) `1

1.	Find the roots of the following equations : (i) `x-1/x=3,x!=0` (ii) `1/(x+4)-1/(x-7)=(11)/(30),x!=-4,7`
Answer» (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.