Question

Use the quadratic formula to solve the quadratic equation 2x2−9x+11=0
2
𝑥
2
−
9
𝑥
+
11
=
0
.

Express its solutions in the form a±bi
𝑎
±
𝑏
𝑖

Answer 1

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:

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