5. Solve \( \frac{3}{m}=-2+\frac{m}{2 m-1} \), giving your answers to 2 decimal places.

To solve the equation \( \frac{3}{m} = -2 + \frac{m}{2m-1} \), first, we'll eliminate the fractions. Let's multiply both sides by \( m(2m-1) \) (assuming \( m \neq 0 \) and \( 2m - 1 \neq 0 \)): \[ 3(2m - 1) = m(-2m + m) \] This simplifies to: \[ 6m - 3 = -2m^2 + m \] Rearranging gives us: \[ 2m^2 + 5m - 3 = 0 \] Now we can use the quadratic formula, \( m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = 5 \), and \( c = -3 \). Calculating the discriminant: \[ b^2 - 4ac = 5^2 - 4(2)(-3) = 25 + 24 = 49 \] Now substituting into the formula: \[ m = \frac{-5 \pm \sqrt{49}}{2(2)} = \frac{-5 \pm 7}{4} \] Calculating the two possible values for \( m \): 1. \( m = \frac{-5 + 7}{4} = \frac{2}{4} = 0.5 \) 2. \( m = \frac{-5 - 7}{4} = \frac{-12}{4} = -3 \) So the solutions to the equation are \( m = 0.5 \) and \( m = -3 \). Finally, rounding these to two decimal places gives: \( m = 0.50 \) and \( m = -3.00 \).