Question

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

Answer 1

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:

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