To solve the quadratic equation \( 2x^2 - 9x + 11 = 0 \) using the quadratic formula, we start with the standard format of a quadratic equation:
\[ ax^2 + bx + c = 0 \]
Here, \( a = 2 \), \( b = -9 \), and \( c = 11 \). The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Now, we will calculate the discriminant \( b^2 - 4ac \):
\[ b^2 - 4ac = (-9)^2 - 4(2)(11) = 81 - 88 = -7 \]
Since the discriminant is negative, this indicates that the solutions will be complex numbers. Now, we can substitute the values into the quadratic formula:
\[ x = \frac{-(-9) \pm \sqrt{-7}}{2(2)} = \frac{9 \pm \sqrt{-7}}{4} \]
We express the square root of the negative number using \( i \):
\[ \sqrt{-7} = i\sqrt{7} \]
Substituting this back into our formula gives:
\[ x = \frac{9 \pm i\sqrt{7}}{4} \]
We can separate the real and imaginary parts:
\[ x = \frac{9}{4} \pm \frac{\sqrt{7}}{4} i \]
Thus, the solutions are:
\[ x = \frac{9}{4} \pm \frac{\sqrt{7}}{4} i \]
In summary, the solutions to the equation \( 2x^2 - 9x + 11 = 0 \) are:
\[ \frac{9}{4} \pm \frac{\sqrt{7}}{4} i \]