How many solutions does the equation (x-p)^2 (x-q) = k
have when k > 0?

To determine how many solutions the equation \((x - p)^2 (x - q) = k\) has when \( k > 0 \), we need to analyze the equation step by step.

### Step 1: Analyze the factors

The left-hand side of the equation is \((x - p)^2 (x - q)\), which is a product of two factors:

1. \((x - p)^2\), a square term, which is always non-negative. It equals zero only when \(x = p\).
2. \((x - q)\), which is a linear term.

### Step 2: Consider critical points

There are two key values for \(x\), namely \(p\) and \(q\), at which the behavior of the equation changes:

- **At \(x = p\)**: \((x - p)^2 = 0\), so the left-hand side becomes zero, regardless of the value of \((x - q)\). Thus, \(x = p\) does **not** satisfy the equation for \(k > 0\).
- **At \(x = q\)**: \((x - p)^2 = (q - p)^2\) and \((x - q) = 0\), so the left-hand side becomes zero, which again does not satisfy the equation for \(k > 0\).

### Step 3: Behavior of the equation for \(x \neq p\)

For \(x \neq p\), \((x - p)^2 > 0\). So, the equation becomes:

\[
(x - p)^2 (x - q) = k
\]

Rearranging:

\[
(x - p)^2 = \frac{k}{x - q}
\]

For \(k > 0\), this equation only holds when \((x - q)\) has the same sign as \(k/(x - q)\). This means the equation behaves differently based on the position of \(x\) relative to \(p\) and \(q\).

### Step 4: Determine number of solutions

The equation involves a quadratic term \((x - p)^2\), which generally leads to two solutions for \(x\), except when \(x = p\). However, for \((x - p)^2\) to equal a positive number, the term \((x - q)\) must also match the positivity of \(k\). This leads to a finite number of intersections (typically two) based on the shape of the quadratic function.

### Conclusion

For \(k > 0\), the equation \((x - p)^2 (x - q) = k\) generally has **two solutions**.

Would you like further details or have any questions?

Here are five related questions to explore:
1. What happens when \(k = 0\)?
2. How many solutions does \((x - p)^2 (x - q) = k\) have when \(k < 0\)?
3. Can we graphically interpret the solution of the equation?
4. What are the solutions if \(p = q\)?
5. How does the number of solutions change if the power of \((x - p)\) is different?

**Tip:** Always check for cases where the equation could be undefined or when key points (like zeros of the factors) affect the solution.

How many solutions does the equation (x-p)^2 (x-q) = k have when k > 0?

Explore how to determine the number of solutions for the equation (x - p)^2 (x - q) = k when k > 0. This problem covers key algebraic concepts such as quadratic equations, factor analysis, and solution counting techniques. Suitable for students in Grades 9-12.

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