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Let's solve the problem step by step.

### Equation: 
\[
x^2 - x + 6 = 0
\]

The discriminant \(\Delta\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula:

\[
\Delta = b^2 - 4ac
\]

In this case, comparing \(x^2 - x + 6 = 0\) with the standard form \(ax^2 + bx + c = 0\):
- \(a = 1\)
- \(b = -1\)
- \(c = 6\)

Now, we substitute these values into the discriminant formula:

\[
\Delta = (-1)^2 - 4(1)(6)
\]
\[
\Delta = 1 - 24
\]
\[
\Delta = -23
\]

### Number and type of solutions:
Since the discriminant is negative (\(\Delta < 0\)), this quadratic equation has **two complex (imaginary) solutions**.

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

Here are 5 related questions to deepen your understanding:
1. What does the discriminant being positive indicate about the solutions?
2. How do you find the exact complex solutions of the equation using the quadratic formula?
3. How would the discriminant change if the constant term were different?
4. What is the significance of the discriminant when it equals zero?
5. How can you visually interpret the discriminant on a graph of a quadratic function?

**Tip:** The discriminant gives a quick way to determine the nature of solutions (real vs. complex) without solving the entire equation!

Find the discriminant of x^2 - x + 6 = 0, and describe the number and type of solutions of the equation.

Solve the quadratic equation x^2 - x + 6 = 0 by finding the discriminant. Learn how to use the discriminant formula to determine the number and type of solutions. This problem covers algebraic concepts suitable for Grades 8-10.

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