what are the roots of the equation 2x^2-11x+16=0 in simplest a + bi form

To find the roots of the quadratic equation \(2x^2 - 11x + 16 = 0\), we can use the quadratic formula:

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

where \(a = 2\), \(b = -11\), and \(c = 16\).

First, let's calculate the discriminant, \(D = b^2 - 4ac\):

\[
D = (-11)^2 - 4(2)(16)
\]

\[
D = 121 - 128
\]

\[
D = -7
\]

Since the discriminant \(D\) is negative, the roots will be complex and can be expressed in the form \(a + bi\).

Now, using the quadratic formula:

\[
x = \frac{-(-11) \pm \sqrt{-7}}{2 \cdot 2}
\]

\[
x = \frac{11 \pm \sqrt{-7}}{4}
\]

We can express \(\sqrt{-7}\) as \(i\sqrt{7}\). Therefore, the roots are:

\[
x = \frac{11 \pm i\sqrt{7}}{4}
\]

So, the roots of the equation \(2x^2 - 11x + 16 = 0\) in the form \(a + bi\) are:

\[
x_1 = \frac{11}{4} + \frac{i\sqrt{7}}{4}, \quad x_2 = \frac{11}{4} - \frac{i\sqrt{7}}{4}
\]

These can also be written as:

\[
x_1 = 2.75 + 0.5i\sqrt{7}, \quad x_2 = 2.75 - 0.5i\sqrt{7}
\]

Would you like more details on any step, or do you have any questions?

Here are some related questions you might be interested in:
1. How do you determine if a quadratic equation has real or complex roots?
2. What is the geometric interpretation of complex roots?
3. How can you verify the roots of a quadratic equation?
4. How are the coefficients of a quadratic equation related to its roots?
5. What are some practical applications of complex numbers in engineering?

**Tip:** When dealing with complex roots, remember that they always occur in conjugate pairs for polynomials with real coefficients.

Learn how to find the roots of the quadratic equation 2x^2 - 11x + 16 = 0 in simplest a + bi form. This problem involves using the quadratic formula and dealing with complex numbers. Suitable for students in Grades 10-12.

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