To solve the inequality \(x^2 - x + 2 \leq 0\), we will first analyze the quadratic expression \(x^2 - x + 2\).

### Step 1: Find the Discriminant

A quadratic equation \(ax^2 + bx + c = 0\) has a discriminant given by:

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

For the expression \(x^2 - x + 2\), we have:
- \(a = 1\)
- \(b = -1\)
- \(c = 2\)

Let's calculate the discriminant:

\[
D = (-1)^2 - 4 \cdot 1 \cdot 2 = 1 - 8 = -7
\]

### Step 2: Analyze the Discriminant

Since the discriminant \(D = -7\) is less than 0, the quadratic expression \(x^2 - x + 2\) has no real roots. This means that the parabola represented by \(y = x^2 - x + 2\) does not intersect the x-axis.

### Step 3: Determine the Direction of the Parabola

The coefficient of \(x^2\) (which is \(a = 1\)) is positive, so the parabola opens upwards. Since the parabola opens upwards and does not intersect the x-axis, the expression \(x^2 - x + 2\) is always greater than 0 for all real values of \(x\).

### Step 4: Conclusion for the Inequality

Given that \(x^2 - x + 2 > 0\) for all \(x \in \mathbb{R}\), the inequality:

\[
x^2 - x + 2 \leq 0
\]

has no solution. 

### Final Answer

There are no real values of \(x\) that satisfy the inequality \(x^2 - x + 2 \leq 0\).

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Would you like more details on the steps, or do you have any questions? Here are some related questions:

1. What happens if the discriminant is zero?
2. How do we solve quadratic inequalities with real roots?
3. Can we visualize the graph of the quadratic equation?
4. What is the effect of the coefficients on the shape of the parabola?
5. How does completing the square help in solving quadratic inequalities?

**Tip:** When the discriminant is negative, the quadratic equation has no real roots, meaning it either remains above or below the x-axis depending on the leading coefficient.

Learn how to solve the quadratic inequality x^2 - x + 2 ≤ 0 using discriminant analysis. Understand why this inequality has no real solution due to the negative discriminant, making it a useful problem for algebra students in Grades 10-12.

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