Chapter SH, Problem 8.16EP

To determine

To find: the solution of the equation and verify it.

Answer to Problem 8.16EP

The solution of the equation is $\text{x=}4\text{ and x=}-1.35$ which is verified.

Explanation of Solution

Given:

  $\text{f}\left(\text{x}\right)\text{=}\frac{\text{2x}}{\text{x-3}}$ .

Concept used:

Here the denominator is taken common by taking lowest common multiple.

To solve the quadratic equation Discriminant method is use here:

If the equation is in the form of ${\text{ax}}^{\text{2}}\text{+bx+c}$ where $\text{a,b and c}$ are constant.

Discriminant formula:

  $\text{x=}\frac{\text{-b±}\sqrt{{\text{b}}^{\text{2}}\text{-4ac}}}{\text{2a}}$ .

Calculation:

  $\frac{\text{3}}{\text{x+4}}\text{+}\frac{\text{1}}{\text{x-1}}\text{=}\frac{\text{17}}{\text{24}}$ …….. $\left(\text{i}\right)$

  $\Rightarrow \frac{\text{3}\left(\text{x-1}\right)\text{+1}\left(\text{x+4}\right)}{\left(\text{x+4}\right)\left(\text{x-1}\right)}\text{=}\frac{\text{17}}{\text{24}}$ .

  $\Rightarrow \frac{\text{3}\left(\text{x-1}\right)\text{+1}\left(\text{x+4}\right)}{\left(\text{x+4}\right)\left(\text{x-1}\right)}\text{=}\frac{\text{17}}{\text{24}}$ .

  $\Rightarrow \frac{\text{3x-3+x+4}}{{\text{x}}^{\text{2}}\text{-x+4x-4}}\text{=}\frac{\text{17}}{\text{24}}$ .

  $\Rightarrow \frac{\text{4x+1}}{{\text{x}}^{\text{2}}\text{+3x-4}}\text{=}\frac{\text{17}}{\text{24}}$ .

By cross multiplying the equation:

  $\text{24}\left(\text{4x+1}\right)=\text{17}\left({\text{x}}^{\text{2}}\text{+3x-4}\right)$ .

  $\Rightarrow {\text{96x+24=17x}}^{\text{2}}\text{+51x-68}$ .

  $\Rightarrow {\text{17x}}^{\text{2}}\text{+51x-68-96x-24}$ .

  ${\text{17x}}^{\text{2}}\text{-45x-92=0}$ .

Here $\text{a=17},\text{b=-45 and c=-92}$ .

  $\text{x=}\frac{\text{-b±}\sqrt{{\text{b}}^{\text{2}}\text{-4ac}}}{\text{2a}}$ .

  $\text{x=}\frac{\text{-}\left(-45\right)\text{±}\sqrt{{\left(-45\right)}^{\text{2}}\text{-4}\left(17\right)\left(-92\right)}}{\text{2}\left(17\right)}$ .

  $\text{x=}\frac{45\text{±}91}{34}$ .

  $\text{x=}\frac{136}{34}\text{ and x=}\frac{-46}{34}$ .

  $\text{x=}4\text{ and x=}-1.35$ .

By putting $\text{x=}4$ in equation $\left(\text{i}\right)$ :

  $\frac{\text{3}}{\text{4+4}}\text{+}\frac{\text{1}}{\text{4-1}}\text{=}\frac{\text{17}}{\text{24}}$ .

  $\Rightarrow \frac{\text{3}}{\text{8}}\text{+}\frac{\text{1}}{\text{3}}\text{=}\frac{\text{17}}{\text{24}}$ .

  $\Rightarrow \frac{9+8}{24}\text{=}\frac{\text{17}}{\text{24}}$ .

  $\Rightarrow \frac{17}{24}\text{=}\frac{\text{17}}{\text{24}}$ verified at $\text{x=}4$ .

By putting $\text{x=}-1.35$ in equation $\left(\text{i}\right)$ :

  $\frac{\text{3}}{\text{-1}\text{.35+4}}\text{+}\frac{\text{1}}{\text{-1}\text{.35-1}}\text{=}\frac{\text{17}}{\text{24}}$ .

  $\frac{\text{3}\left(\text{-2}\text{.35}\right)+\text{2}\text{.65}}{\text{2}\text{.65}\left(\text{-2}\text{.35}\right)}\text{=}\frac{\text{17}}{\text{24}}$ .

  $\text{0}\text{.708=0}\text{.708}$ verified at $\text{x=}-1.35$ .

Hence, the solution of the equation is $\text{x=}4\text{ and x=}-1.35$ which is verified.