To solve the equation
\[ \frac{2}{x} - \frac{3x}{x+3} = \frac{x}{x+3}, \]
we can first eliminate the fractions by multiplying both sides of the equation by \( x(x + 3) \), which is the common denominator. This eliminates the fractions, allowing us to simplify the equation:
\[ (x + 3)(2) - 3x(x) = x(x). \]
Expanding both sides gives:
\[ 2x + 6 - 3x^2 = x^2. \]
Now, rearranging the equation brings all terms to one side:
\[ -3x^2 - x^2 + 2x + 6 = 0, \]
which simplifies to:
\[ -4x^2 + 2x + 6 = 0. \]
Multiplying throughout by -1 to make the leading coefficient positive:
\[ 4x^2 - 2x - 6 = 0. \]
Next, we can simplify this equation by dividing all terms by 2:
\[ 2x^2 - x - 3 = 0. \]
Now we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = -1 \), and \( c = -3 \):
\[ b^2 - 4ac = (-1)^2 - 4 \cdot 2 \cdot (-3) = 1 + 24 = 25. \]
Now substituting back into the quadratic formula:
\[ x = \frac{-(-1) \pm \sqrt{25}}{2 \cdot 2} = \frac{1 \pm 5}{4}. \]
This gives us two solutions:
- \( x = \frac{6}{4} = \frac{3}{2} \)
- \( x = \frac{-4}{4} = -1 \)
Thus the solutions to the equation are \( x = -1 \) and \( x = \frac{3}{2} \).
Final answer:
−1 and \(\frac{3}{2}\).