(2) \( x^{2}+m x+9=0 \)

To solve the quadratic equation \( x^{2} + m x + 9 = 0 \), we can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] In this equation, \( a = 1 \), \( b = m \), and \( c = 9 \). Now, let's calculate the discriminant \( D \) first, which is given by: \[ D = b^2 - 4ac \] Substituting the values of \( a \), \( b \), and \( c \): \[ D = m^2 - 4 \cdot 1 \cdot 9 = m^2 - 36 \] Next, we can substitute \( D \) back into the quadratic formula to find the solutions for \( x \): \[ x = \frac{-m \pm \sqrt{m^2 - 36}}{2} \] Now, let's summarize the solutions: 1. If \( m^2 - 36 > 0 \), there are two distinct real solutions. 2. If \( m^2 - 36 = 0 \), there is one real solution (a repeated root). 3. If \( m^2 - 36 < 0 \), there are no real solutions (the solutions are complex). Would you like to explore specific values for \( m \) or further analyze the solutions?