To solve the quadratic equation \(2x^2 - 9x + 11 = 0\) using the quadratic formula, we first identify the coefficients \(a\), \(b\), and \(c\) in the standard form of the quadratic equation \(ax^2 + bx + c = 0\).
Here:
- \(a = 2\)
- \(b = -9\)
- \(c = 11\)
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Now, we will calculate \(b^2 - 4ac\):
\[ b^2 = (-9)^2 = 81 \] \[ 4ac = 4 \cdot 2 \cdot 11 = 88 \] \[ b^2 - 4ac = 81 - 88 = -7 \]
Since \(b^2 - 4ac\) is negative, we have no real solutions, but we can find complex solutions.
Now substituting the values into the quadratic formula:
\[ x = \frac{-(-9) \pm \sqrt{-7}}{2 \cdot 2} = \frac{9 \pm \sqrt{-7}}{4} \]
Now we express \(\sqrt{-7}\) in terms of \(i\):
\[ \sqrt{-7} = i\sqrt{7} \]
Substituting this back, we get:
\[ x = \frac{9 \pm i\sqrt{7}}{4} \]
Thus, the solutions to the equation \(2x^2 - 9x + 11 = 0\) are:
\[ x = \frac{9 + i\sqrt{7}}{4} \quad \text{and} \quad x = \frac{9 - i\sqrt{7}}{4} \]