Question

Find the roots of the following equation $\frac{1}{x+4}-\frac{1}{x-7}=\frac{11}{30}$.

Answer 1

Step 1: Estimate the value of $a,b$ and $c$

Given equation is

$\frac{1}{x+4}-\frac{1}{x-7}=\frac{11}{30}$

$\Rightarrow$ $\frac{x-7-x-4}{\left(x+4\right)\left(x-7\right)}=\frac{11}{30}$

$\Rightarrow$$\frac{-11}{x^2-7x+4x-28}=\frac{11}{30}$

$\Rightarrow$ $x^2-3x-28=-30$

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

According to the Sridharacharya formula, for a given equation $a{x}^2+bx+c$, the roots of this equation are given by

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

From the given equation we have,

$a=1,b=-3,c=2$

Step 2: Determine the roots of the equation

$\therefore$The roots of the equation are,

$x=\frac{-\left(-3\right)\pm \sqrt{{\left(-3\right)}^2-4\times 1\times 2}}{2\times 1}$

$=\frac{3\pm \sqrt{9-8}}{2}$

$=\frac{3\pm 1}{2}$

$=\frac{4}{2},\frac{2}{2}$

$=2,1$

Hence, the roots of the equation are $2$ and $1$.