What is the solution to (x + 2)(x – 3)2 > 0?

The inequality is \((x + 2)(x - 3)^2 > 0\). To solve this, follow these steps:

### Step 1: Find the critical points.
The critical points are the values of \(x\) where the expression equals zero. Set each factor equal to zero:

1. \(x + 2 = 0\) gives \(x = -2\).
2. \((x - 3)^2 = 0\) gives \(x = 3\).

### Step 2: Analyze the sign of the expression in different intervals.
The critical points divide the number line into three intervals: \((- \infty, -2)\), \((-2, 3)\), and \((3, \infty)\). We will determine the sign of the expression in each of these intervals by choosing test points.

1. **Interval \( (- \infty, -2) \):** Choose \(x = -3\).
   \[
   (x + 2)(x - 3)^2 = (-3 + 2)(-3 - 3)^2 = (-1)(-6)^2 = (-1)(36) = -36
   \]
   The expression is negative in this interval.

2. **Interval \( (-2, 3) \):** Choose \(x = 0\).
   \[
   (x + 2)(x - 3)^2 = (0 + 2)(0 - 3)^2 = (2)(-3)^2 = (2)(9) = 18
   \]
   The expression is positive in this interval.

3. **Interval \( (3, \infty) \):** Choose \(x = 4\).
   \[
   (x + 2)(x - 3)^2 = (4 + 2)(4 - 3)^2 = (6)(1)^2 = 6
   \]
   The expression is positive in this interval.

### Step 3: Consider the critical points.
- At \(x = -2\), \((x + 2)(x - 3)^2 = 0\), so the expression is zero.
- At \(x = 3\), \((x + 2)(x - 3)^2 = 0\), because \((x - 3)^2 = 0\) at this point.

### Step 4: Determine the solution.
We are looking for where the expression is greater than 0 (positive). From the sign analysis:

- The expression is negative for \(x < -2\).
- The expression is positive for \(-2 < x < 3\) and \(x > 3\).
- At \(x = -2\) and \(x = 3\), the expression is zero, which does not satisfy the inequality.

Thus, the solution is:
\[
(-2, 3) \cup (3, \infty)
\]

Would you like any additional details or explanations?

### Related questions:
1. How do you solve polynomial inequalities in general?
2. What are critical points, and why are they important in inequalities?
3. How does squaring a factor affect the solution of inequalities?
4. What is the significance of testing points in different intervals for inequalities?
5. How do you graph solutions to polynomial inequalities?

### Tip:
Always check how factors raised to even powers behave in inequalities—they are non-negative and may affect the solution boundaries.

What is the solution to (x + 2)(x – 3)^2 > 0?

Learn how to solve the inequality (x + 2)(x – 3)^2 > 0 with a detailed step-by-step breakdown. This solution covers key algebraic concepts like critical points, interval analysis, and how to handle squared terms in inequalities. Suitable for students in Grades 10-12.

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