M²+6m-19=0

To solve the quadratic equation \( m^2 + 6m - 19 = 0 \), we can use the quadratic formula: \[ m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1 \), \( b = 6 \), and \( c = -19 \). First, we calculate the discriminant (\( b^2 - 4ac \)): \[ b^2 - 4ac = 6^2 - 4 \cdot 1 \cdot (-19) = 36 + 76 = 112 \] Now, we can substitute the values into the quadratic formula: \[ m = \frac{-6 \pm \sqrt{112}}{2 \cdot 1} \] Next, simplify \( \sqrt{112} \): \[ \sqrt{112} = \sqrt{16 \cdot 7} = 4\sqrt{7} \] Now substitute back into the equation: \[ m = \frac{-6 \pm 4\sqrt{7}}{2} \] This simplifies to: \[ m = \frac{-6}{2} \pm \frac{4\sqrt{7}}{2} \] \[ m = -3 \pm 2\sqrt{7} \] Thus, the two solutions for the equation \( m^2 + 6m - 19 = 0 \) are: \[ m_1 = -3 + 2\sqrt{7} \quad \text{and} \quad m_2 = -3 - 2\sqrt{7} \]