(12)/(x-2)+(0,5)/(1)-(15)/(x)=0

To solve the equation \( \frac{12}{x-2} + \frac{0.5}{1} - \frac{15}{x} = 0 \), we need to find a common denominator for the fractions. The common denominator for \( x-2 \), \( 1 \), and \( x \) is \( (x-2) \cdot x \cdot 1 \).<br /><br />Rewriting the equation with the common denominator, we get:<br /><br />\[ \frac{12x}{(x-2)x} + \frac{0.5(x--2)x} - \frac{15(x-2)}{(x-2)x} = 0 \]<br /><br />Now, we can combine the fractions:<br /><br />\[ \frac{12x + 0.5(x-2)x - 15(x-2)}{(x-2)x} = 0 \]<br /><br />Since the denominator cannot be zero, we can multiply both sides of the equation by \( (x-2)x \) to eliminate the denominator:<br /><br />\[ 12x + 0.5(x-2)x - 15(x-2) = 0 \]<br /><br />Expanding and simplifying the equation, we get:<br /><br />\[ 12x + 0.5x^2 - 2x - 15x + 30 = 0 \]<br /><br />Combining like terms, we have:<br /><br />\[ 0.5x^2 - 5x + 30 = 0 \]<br /><br />To solve this quadratic equation, we can use the quadratic formula:<br /><br />\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]<br /><br />In this case, \( a = 0.5 \), \( b = -5 \), and \( c = 30 \). Plugging these values into the quadratic formula, we get:<br /><br />\[ x = \frac{5 \pm \sqrt{25 - 60}}{1} \]<br /><br />\[ x = \frac{5 \pm \sqrt{-35}}{1} \]<br /><br />Since the discriminant (\( b^2 - 4ac \)) is negative, there are no real solutions for this equation. Therefore, the equation \( \frac{12}{x-2} + \frac{0.5}{1} - \frac{15}{x} = 0 \) has no real solutions.