To solve the quadratic equation \(2x^2 + 9x + 11 = 0\) using the quadratic formula, we start by identifying the coefficients:
- \(a = 2\)
- \(b = 9\)
- \(c = 11\)
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
First, we calculate the discriminant (\(b^2 - 4ac\)):
\[ b^2 - 4ac = 9^2 - 4 \cdot 2 \cdot 11 = 81 - 88 = -7 \]
Since the discriminant is negative, the solutions will be complex. Now we can substitute the values into the quadratic formula:
\[ x = \frac{-9 \pm \sqrt{-7}}{2 \cdot 2} = \frac{-9 \pm \sqrt{7}i}{4} \]
So the solutions in the form \(a \pm bi\) are:
\[ x = -\frac{9}{4} \pm \frac{\sqrt{7}}{4}i \]
Thus, the final answers are:
\[ x = -\frac{9}{4} + \frac{\sqrt{7}}{4}i \quad \text{and} \quad x = -\frac{9}{4} - \frac{\sqrt{7}}{4}i \]