Question

Find the roots of the following

$1 \div s + 4 \: - \: 1 \div s - 7 = 11 \div 30$
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Answer 1

Answer:

The roots of the given quadratic equation are

$\displaystyle{\boxed{\red{\sf\:s\:=\:2}}\:\sf\quad\:OR\:\quad\:\boxed{\red{\sf\:s\:=\:1}}}$

Step-by-step-explanation:

We have given a quadratic equation.

We have to find the roots of the equation.

The given quadratic equation is

$\displaystyle{\sf\:\dfrac{1}{s\:+\:4}\:-\:\dfrac{1}{s\ -\ 7}\ =\ \dfrac{11}{30}}$

$\displaystyle{\implies\sf\:\dfrac{(\:s\:-\:7\:)\:-\:(\:s\:+\:4\:)}{(\:s\:+\ 4\ )\:(\:s\ -\ 7\ )}\ =\ \dfrac{11}{30}}$

$\displaystyle{\implies\sf\:\dfrac{\cancel{s}\:-\:7\:-\:\cancel{s}\:-\:4}{s\:(\:s\:-\:7\:)\:+\:4\:(\:s\:-\:7\:)}\:=\:\dfrac{11}{30}}$

$\displaystyle{\implies\sf\:\dfrac{-\:11}{s^2\:-\:7s\:+\:4s\:-\:28}\:=\:\dfrac{11}{30}}$

$\displaystyle{\implies\sf\:\dfrac{-\:11}{s^2\:-\:3s\:-\:28}\:=\:\dfrac{11}{30}}$

$\displaystyle{\implies\sf\:30\:\times\:(\:-\:11\:)\:=\:11\:\times\:(\:s^2\:-\:3s\:-\:28\:)}$

$\displaystyle{\implies\sf\:-\:330\:=\:11\:\times\:(\:s^2\:-\:3s\:-\:28\:)}$

$\displaystyle{\implies\sf\:s^2\:-\:3s\:-\:28\:=\:\dfrac{-\:\cancel{330}}{\cancel{11}}}$

$\displaystyle{\implies\sf\:s^2\:-\:3s\:-\:28\:=\:-\:30}$

$\displaystyle{\implies\sf\:s^2\:-\:3s\:-\:28\:+\:30\:=\:0}$

$\displaystyle{\implies\sf\:s^2\:-\:3s\:+\:2\:=\:0}$

$\displaystyle{\implies\sf\:s^2\:-\:2s\:-\:s\:+\:2\:=\:0}$

$\displaystyle{\implies\sf\:s\:(\:s\:-\:2\:)\:-\:1\:(\:s\:-\:2\:)\:=\:0}$

$\displaystyle{\implies\sf\:(\:s\:-\:2\:)\:(\:s\:-\:1\:)\:=\:0}$

$\displaystyle{\implies\sf\:(\:s\:-\:2\:)\:=\:0\:\quad\:OR\:\quad\:(\:s\:-\:1\:)\:=\:0}$

$\displaystyle{\implies\sf\:s\:-\:2\:=\:0\:\quad\:OR\:\quad\:s\:-\:1\:=\:0}$

$\displaystyle{\implies\underline{\boxed{\red{\sf\:s\:=\:2}}}\:\sf\quad\:OR\:\quad\:\underline{\boxed{\red{\sf\:s\:=\:1}}}}$