Question

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

Express its solutions in the form a±bi
.

(1 point)

Answer 1

To solve the quadratic equation \( 2x^2 - 9x + 11 = 0 \) using the quadratic formula, we start with the standard format of a quadratic equation:

\[
ax^2 + bx + c = 0
\]

Here, \( a = 2 \), \( b = -9 \), and \( c = 11 \). The quadratic formula is given by:

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

Now, we will calculate the discriminant \( b^2 - 4ac \):

\[
b^2 - 4ac = (-9)^2 - 4(2)(11) = 81 - 88 = -7
\]

Since the discriminant is negative, this indicates that the solutions will be complex numbers. Now, we can substitute the values into the quadratic formula:

\[
x = \frac{-(-9) \pm \sqrt{-7}}{2(2)} = \frac{9 \pm \sqrt{-7}}{4}
\]

We express the square root of the negative number using \( i \):

\[
\sqrt{-7} = i\sqrt{7}
\]

Substituting this back into our formula gives:

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

We can separate the real and imaginary parts:

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

Thus, the solutions are:

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

In summary, the solutions to the equation \( 2x^2 - 9x + 11 = 0 \) are:

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