Question

solve: 1/x-4-1/x-7=1/30. Quadratic equation

Answer 1

Answer: The answer is $x=\dfrac{11\pm i\sqrt{351}}{2}.$

Step-by-step explanation:  The given quadratic equation to solve is

$\dfrac{1}{x-4}-\dfrac{1}{x-7}=\dfrac{1}{30}.$

Recall that the solution of the quadratic equation $ax^2+bx+c=0,~a\neq 0$ is given by

$x=\dfrac{-b+\sqrt{b^2-4ac}}{2a}.$

We have

$\dfrac{1}{x-4}-\dfrac{1}{x-7}=\dfrac{1}{30}\\\\\\\Rightarrow \dfrac{x-7-x+4}{(x-4)(x-7)}=\dfrac{1}{30}\\\\\\\Rightarrow \dfrac{-3}{x^2-11x+28}=\dfrac{1}{30}\\\\\\\Rightarrow x^2-11x+28=-90\\\\\Rightarrow x^2-11x+118=0\\\\\Rightarrow x=\dfrac{-(-11)\pm\sqrt{(-11)^2-4\times 1\times 118}}{2\times 1}\\\\\\\Rightarrow x=\dfrac{11\pm\sqrt{121-472}}{2}\\\\\\\Rightarrow x=\dfrac{11\pm\sqrt{-351}}{2}\\\\\\\Rightarrow x=\dfrac{11\pm i\sqrt{351}}{2}.$

Hence solved.