To solve the quadratic equation \( 2x^2 - 9x + 11 = 0 \) using the quadratic formula, we start with the formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In your equation, the coefficients are:
- \( a = 2 \)
- \( b = -9 \)
- \( c = 11 \)
First, we calculate the discriminant \( b^2 - 4ac \):
\[ b^2 = (-9)^2 = 81 \] \[ 4ac = 4 \cdot 2 \cdot 11 = 88 \] \[ b^2 - 4ac = 81 - 88 = -7 \]
Since the discriminant is negative (\(-7\)), the solutions will involve complex numbers. Now, we can substitute the values into the quadratic formula:
\[ x = \frac{-(-9) \pm \sqrt{-7}}{2 \cdot 2} \] \[ x = \frac{9 \pm \sqrt{-7}}{4} \]
We can express \(\sqrt{-7}\) as \(i\sqrt{7}\):
\[ x = \frac{9 \pm i\sqrt{7}}{4} \]
This can be separated into real and imaginary parts:
\[ x = \frac{9}{4} \pm \frac{\sqrt{7}}{4}i \]
Thus, the solutions are:
\[ x = \frac{9}{4} + \frac{\sqrt{7}}{4}i \quad \text{and} \quad x = \frac{9}{4} - \frac{\sqrt{7}}{4}i \]
In summary, the solutions expressed in the form \( a \pm bi \) are:
\[ \frac{9}{4} \pm \frac{\sqrt{7}}{4}i \]