Question

solve \\(\frac{x + 5}{x} = \frac{2x - 3}{8}\\).
a. \\(-\frac{5}{2}\\) or 8
b. \\(\frac{5}{2}\\) or 8
c. \\(-\frac{3}{2}\\) or -5
d. \\(\frac{3}{2}\\) or -5

Explanation:

Step1: Cross-multiply to eliminate denominators

$8(x + 5) = x(2x - 3)$

Step2: Expand both sides

$8x + 40 = 2x^2 - 3x$

Step3: Rearrange into standard quadratic form

$2x^2 - 11x - 40 = 0$

Step4: Solve quadratic equation

Using quadratic formula $x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$ where $a=2$, $b=-11$, $c=-40$:
$x = \frac{11\pm\sqrt{121 + 320}}{4} = \frac{11\pm21}{4}$
$x = \frac{32}{4}=8$ or $x=\frac{-10}{4}=-\frac{5}{2}$

Answer:

A. $-\frac{5}{2}$ or 8