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The image shows a mathematical problem asking to find the roots of the function \( f(x) = 2x^2 - 3x + 5 \).

### To solve for the roots, we need to use the **quadratic formula**:

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

For the quadratic equation \( ax^2 + bx + c = 0 \), the coefficients are:
- \( a = 2 \)
- \( b = -3 \)
- \( c = 5 \)

Let's plug in these values into the quadratic formula:

\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(5)}}{2(2)}
\]
\[
x = \frac{3 \pm \sqrt{9 - 40}}{4}
\]
\[
x = \frac{3 \pm \sqrt{-31}}{4}
\]

Since \( \sqrt{-31} \) is an imaginary number, we write it as \( \sqrt{31}i \), where \( i \) is the imaginary unit. Thus, the solutions are:

\[
x = \frac{3 \pm \sqrt{31}i}{4}
\]

### The roots are:
\[
x = \frac{3 + \sqrt{31}i}{4} \quad \text{and} \quad x = \frac{3 - \sqrt{31}i}{4}
\]

These are complex roots because the discriminant \( b^2 - 4ac \) is negative.

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Would you like further details or have any questions?

Here are 5 related questions you might find useful:
1. How do you interpret the discriminant in a quadratic equation?
2. Can you apply the quadratic formula to equations with real solutions?
3. How do you simplify complex numbers in solutions?
4. What are the steps for completing the square to solve quadratic equations?
5. How do you factor quadratic expressions when the discriminant is positive?

**Tip:** Always check the discriminant \( b^2 - 4ac \) before solving a quadratic equation to determine the nature of the roots (real or complex).

Find the roots of the function f(x) = 2x^2 - 3x + 5.

Explore how to find the roots of the quadratic equation 2x^2 - 3x + 5 using the quadratic formula. This problem includes complex solutions due to a negative discriminant, making it suitable for Grades 9-11 students learning advanced algebra concepts.

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