Let's solve the inequality \( x^2 - 5x + 6 < 2 \).

### Step 1: Rearrange the inequality
First, we need to move all terms to one side of the inequality to simplify it:

\[
x^2 - 5x + 6 - 2 < 0
\]

This simplifies to:

\[
x^2 - 5x + 4 < 0
\]

### Step 2: Solve the corresponding equation
Next, solve the quadratic equation \( x^2 - 5x + 4 = 0 \) to find the critical points. We can use the quadratic formula:

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

where \( a = 1 \), \( b = -5 \), and \( c = 4 \).

\[
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(4)}}{2(1)}
\]

\[
x = \frac{5 \pm \sqrt{25 - 16}}{2}
\]

\[
x = \frac{5 \pm \sqrt{9}}{2}
\]

\[
x = \frac{5 \pm 3}{2}
\]

So, the solutions are:

\[
x = \frac{5 + 3}{2} = 4 \quad \text{and} \quad x = \frac{5 - 3}{2} = 1
\]

### Step 3: Determine the intervals
The roots divide the number line into three intervals: \( (-\infty, 1) \), \( (1, 4) \), and \( (4, \infty) \).

### Step 4: Test the intervals
We need to determine where the quadratic expression \( x^2 - 5x + 4 \) is negative.

1. **For the interval \( (-\infty, 1) \)**: Choose \( x = 0 \),
   \[
   0^2 - 5(0) + 4 = 4 > 0
   \]
   So, it’s positive in this interval.

2. **For the interval \( (1, 4) \)**: Choose \( x = 2 \),
   \[
   2^2 - 5(2) + 4 = 4 - 10 + 4 = -2 < 0
   \]
   So, it’s negative in this interval.

3. **For the interval \( (4, \infty) \)**: Choose \( x = 5 \),
   \[
   5^2 - 5(5) + 4 = 25 - 25 + 4 = 4 > 0
   \]
   So, it’s positive in this interval.

### Step 5: Conclusion
The solution to the inequality \( x^2 - 5x + 4 < 0 \) is:

\[
\boxed{1 < x < 4}
\]

Do you have any questions or would you like more details?

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Here are five related questions you might find interesting:

1. How do you solve quadratic inequalities using the factorization method?
2. What is the significance of the discriminant in quadratic equations?
3. How do you graph quadratic inequalities on a number line?
4. How do you solve absolute value inequalities?
5. What are the differences between solving quadratic equations and quadratic inequalities?

**Tip:** Always test the intervals determined by the roots to see where the inequality holds true.

Learn how to solve the quadratic inequality x^2 - 5x + 6 < 2 with a detailed step-by-step solution. This problem involves solving quadratic inequalities using the quadratic formula and testing intervals. Suitable for students in Grades 9-12.

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