Question

Use the quadratic formula to solve the quadratic equation 2x2−9x+11=0 Express its solutions in the form a±bi .

Answer 1

To solve the quadratic equation \(2x^2 - 9x + 11 = 0\) using the quadratic formula, we identify the coefficients \(a\), \(b\), and \(c\):

- \(a = 2\)
- \(b = -9\)
- \(c = 11\)

The quadratic formula is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

First, we need to compute 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 be complex. Now, substituting 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 \(i\sqrt{7}\) (where \(i\) is the imaginary unit):

\[
x = \frac{9 \pm i\sqrt{7}}{4}
\]

This can be separated into real and imaginary parts:

\[
x = \frac{9}{4} \pm \frac{\sqrt{7}}{4}i
\]

Thus, the solutions in the form \(a \pm bi\) are:

\[
x = \frac{9}{4} \pm \frac{\sqrt{7}}{4}i
\]

So the final solutions are:

\[
x = \frac{9}{4} + \frac{\sqrt{7}}{4}i \quad \text{and} \quad x = \frac{9}{4} - \frac{\sqrt{7}}{4}i
\]