Question

Solve each of the following quadratic equations:
(i) $\frac{1}{x+1}+\frac{2}{x+2}=\frac{5}{x+4},x\ne -1,-2,-4$
(ii) $\frac{1}{x+1}+\frac{3}{5x+1}=\frac{5}{x+4},x\ne -1\frac{1}{5},-4$

Answer 1

(i) $\frac{1}{x+1}+\frac{2}{x+2}=\frac{5}{x+4}$

$\Rightarrow \frac{x+2+2x+2}{(x+1)(x+2)}=\frac{5}{x+4}$

$\Rightarrow \frac{3x+4}{x^2+3x+2}=\frac{5}{x+4}$

$\Rightarrow 5x^2+15x+10=3x^2+16x+16$

$\Rightarrow 2x^2-x-6=0$

$\Rightarrow 2x^2-4x+3x-6=0$

$\Rightarrow 2x(x-2)+3(x-2)=0$

$\Rightarrow (2x+3)(x-2)=0$

$\Rightarrow 2x+3=0orx-2=0$

$\Rightarrow x=\frac{-3}{2}orx=2$

(ii) $\frac{1}{x+1}+\frac{3}{5x+1}=\frac{5}{x+4}\frac{5x+1+3x+3}{(x+1)(5x+1)}=\frac{5}{x+4}\frac{8x+4}{5x^2+6x+1}=\frac{5}{x+4}(8x+4)(x+4)=25x^2+30x+58x^2+36x+16=25x^2+30x+517x^2-6x-11=017x^2-17x+11x-11=017x(x-1)+11(x-1)=0(17x+11)(x-1)=017x+11=0orx-1=0x=\frac{11}{17}or1$