To solve the quadratic equation \(2x^2 - 9x + 11 = 0\) using the quadratic formula, we start with the formula itself:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our equation, the coefficients are:
- \(a = 2\)
- \(b = -9\)
- \(c = 11\)
Let's first calculate the discriminant \(b^2 - 4ac\):
\[ b^2 = (-9)^2 = 81 \] \[ 4ac = 4 \times 2 \times 11 = 88 \] \[ \text{Discriminant} = b^2 - 4ac = 81 - 88 = -7 \]
Since the discriminant is negative (\(-7\)), this means the solutions will involve imaginary numbers.
Next, we substitute the values into the quadratic formula:
\[ x = \frac{-(-9) \pm \sqrt{-7}}{2 \cdot 2} \] \[ x = \frac{9 \pm \sqrt{-7}}{4} \]
Now, let's express \( \sqrt{-7} \) in terms of \(i\):
\[ \sqrt{-7} = i\sqrt{7} \]
Now we substitute this back into the formula:
\[ x = \frac{9 \pm i\sqrt{7}}{4} \]
We can separate this into real and imaginary parts:
\[ x = \frac{9}{4} \pm \frac{i\sqrt{7}}{4} \]
Thus, the final solutions in the form \(a \pm bi\) are:
\[ x = \frac{9}{4} \pm \frac{\sqrt{7}}{4} i \]
So, the solutions to the equation \(2x^2 - 9x + 11 = 0\) are:
\[ x = \frac{9}{4} + \frac{\sqrt{7}}{4} i \quad \text{and} \quad x = \frac{9}{4} - \frac{\sqrt{7}}{4} i \]