Question

Solve each of the following quadratic equations:
(i) $\frac{1}{x-1}-\frac{1}{x+5}=\frac{6}{7},x\ne 1,-5$
(ii) $\frac{1}{2x-3}-\frac{1}{x-5}=1\frac{1}{9},x\ne \frac{3}{2},5$

Answer 1

(i) $\frac{1}{x-1}-\frac{1}{x+5}=\frac{6}{7}$

$\frac{(x+5-x+1)}{(x-1)(x+5)}=\frac{6}{7}$

$\frac{6}{(x-1)(x+5)}=\frac{6}{7}$

$\frac{6\times 7}{6}=x^2+5x-x-5$

$7=x^2+4x-5$

$x^2+4x-12=0$

$x^2+(6-2)x-12=0$

$x^2+6x-2x-12=0$

$x(x+6)-2(x+6)=0$

$(x+6)(x-2)=0$

When
$x+6=0\Rightarrow x=-6$

and
$x-2=0\Rightarrow x=2$

x= 2 and -6

(ii) $\frac{1}{2x-3}-\frac{1}{x-5}=1\frac{1}{9}$

$\frac{x-5+2x-3}{(2x-3)(x-5)}=\frac{10}{9}$

$\Rightarrow \frac{3x-8}{2x^2-10x-3x+15}=\frac{10}{9}$

$\Rightarrow \frac{3x-8}{2x^2-13x+15}=\frac{10}{9}$

Cross multiply to get,

$10(2x^2-13x+15)=9(3x-8)$

$\Rightarrow 20x^2-130x+150=27x-72$

$\Rightarrow 20x^2-157x+222=0$

Factorise and solve,

$20x^2-120x-37x+222=0\Rightarrow 20x(x-6)-37(x-6)=0$

$\Rightarrow (x-6)(20x-37)=0$

$\Rightarrow x=6,\frac{37}{20}$

values of x are 6 and $\frac{37}{20}$