Answer
Conditional equation.
$x=\frac{4}{7}$
Work Step by Step
Given equation,
$\frac{4}{x+2} + \frac{3}{x} = \frac{10}{x^{2}+2x} $
$x=0 $ or $ x=-2$ makes the denominator zero, so,
$\frac{4}{x+2} + \frac{3}{x} = \frac{10}{x^{2}+2x} ; x\ne0,-2$
Taking LCD,
$\frac{4x+3(x+2)}{x(x+2)} = \frac{10}{x^{2}+2x} ; x\ne0,-2$
$\frac{4x+3x+6}{x(x+2)} = \frac{10}{x^{2}+2x} ; x\ne0,-2$
$\frac{7x+6}{x^{2}+2x} = \frac{10}{x^{2}+2x} ; x\ne0,-2$
Multiply both sides by $x^{2}+2x$
$(x^{2}+2x)\frac{7x+6}{x^{2}+2x} = (x^{2}+2x)\frac{10}{x^{2}+2x} ; x\ne0,-2$
$7x+6=10 ; x\ne0,-2$
$7x=10-6 ; x\ne0,-2$
$7x=4; x\ne0,-2$
$x=\frac{4}{7}; x\ne0,-2$
This equation has one solution, so it is a conditional equation.
