Find the solution set of `(x^(2))/(x-3)gt0`.

Answer

To solve the inequality \(\frac{x^2}{x-3} > 0\), we will follow these steps:

### Step 1: Identify the critical points
The critical points occur when the numerator or the denominator is zero.

- The numerator \(x^2 = 0\) gives us \(x = 0\).
- The denominator \(x - 3 = 0\) gives us \(x = 3\).

### Step 2: Determine the intervals
The critical points divide the number line into intervals. We will consider the intervals:
1. \( (-\infty, 0) \)
2. \( (0, 3) \)
3. \( (3, \infty) \)

### Step 3: Test each interval
We will test a point from each interval to determine where the inequality holds.

1. **Interval \( (-\infty, 0) \)**: Choose \(x = -1\)
\[ \frac{(-1)^2}{-1 - 3} = \frac{1}{-4} < 0 \quad \text{(not a solution)} \]

2. **Interval \( (0, 3) \)**: Choose \(x = 1\)
\[ \frac{1^2}{1 - 3} = \frac{1}{-2} < 0 \quad \text{(not a solution)} \]

3. **Interval \( (3, \infty) \)**: Choose \(x = 4\)
\[ \frac{4^2}{4 - 3} = \frac{16}{1} > 0 \quad \text{(solution)} \]

### Step 4: Consider the critical points
- At \(x = 0\), \(\frac{0^2}{0 - 3} = 0\) (not included in the solution set since we need \(> 0\)).
- At \(x = 3\), the expression is undefined (not included in the solution set).

### Step 5: Write the solution set
The solution set is \(x > 3\).

Thus, the final answer is:
\[ \text{Solution set: } (3, \infty) \]

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