https://file.aiquickdraw.com/mathbot/file/172545700060789xjmfkq.jpg

The expression in the image is:

\[ x^2 - 3x + 10 \]

This is a quadratic expression, but it cannot be factored into real numbers since its discriminant is negative. Let's analyze it step by step:

1. **Standard Form**: The expression is already in the standard quadratic form \( ax^2 + bx + c \), where:
   - \( a = 1 \)
   - \( b = -3 \)
   - \( c = 10 \)

2. **Discriminant**: The discriminant \( \Delta \) is given by:
   \[
   \Delta = b^2 - 4ac
   \]
   Substituting the values:
   \[
   \Delta = (-3)^2 - 4(1)(10) = 9 - 40 = -31
   \]
   Since the discriminant is negative (\( \Delta = -31 \)), the quadratic does not have real roots, only complex ones.

3. **Complex Roots**: The roots of the quadratic equation can be found using the quadratic formula:
   \[
   x = \frac{-b \pm \sqrt{\Delta}}{2a}
   \]
   Substituting the values:
   \[
   x = \frac{3 \pm \sqrt{-31}}{2} = \frac{3 \pm i\sqrt{31}}{2}
   \]

   The roots are complex:
   \[
   x_1 = \frac{3 + i\sqrt{31}}{2}, \quad x_2 = \frac{3 - i\sqrt{31}}{2}
   \]

Would you like more details on this solution or have any questions?

Here are 5 related questions for further exploration:

1. How do you determine the nature of the roots of any quadratic equation?
2. What is the significance of the discriminant in determining the roots?
3. How can you solve quadratic equations with complex roots?
4. What are the steps to completing the square for a quadratic expression?
5. How would you graph the quadratic function \( y = x^2 - 3x + 10 \) on the complex plane?

**Tip**: Always check the discriminant first to quickly determine the nature of the roots (real or complex) before attempting to factorize or use the quadratic formula.

Solve the quadratic expression x^2 - 3x + 10.

Explore how to solve the quadratic expression x^2 - 3x + 10. This problem includes a step-by-step breakdown of the discriminant, reveals the nature of the roots, and provides insight into complex number solutions. Perfect for students in Grades 9-12.

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