https://file.aiquickdraw.com/mathbot/file/172537470452841h9aqwv.jpg

The equation provided in the image is:

\[
\frac{z^2 - 16z}{z} = -32
\]

Let's solve this step by step.

### Step 1: Simplify the equation
First, simplify the left-hand side of the equation:

\[
\frac{z^2 - 16z}{z} = z - 16
\]

So, the equation becomes:

\[
z - 16 = -32
\]

### Step 2: Solve for \( z \)
Now, solve for \( z \) by adding 16 to both sides of the equation:

\[
z = -32 + 16
\]

\[
z = -16
\]

So, the solution is:

\[
z = -16
\]

If you need further details or have any questions, feel free to ask!

### Related Questions:
1. What if the equation was \(\frac{z^2 - 16z + 64}{z} = -32\)?
2. How would you solve the equation if it was \(\frac{z^3 - 16z^2}{z} = -32\)?
3. What happens if the equation had a different constant, such as \(-40\) instead of \(-32\)?
4. How do you handle equations where \(z\) cannot be factored easily?
5. Can you solve this equation graphically?

### Tip:
When dealing with equations that involve fractions, try to simplify the fraction first before solving the equation. This often makes the problem more manageable.

Solve for z: \( \frac{z^2 - 16z}{z} = -32 \)

Learn how to solve the equation \( \frac{z^2 - 16z}{z} = -32 \) by simplifying algebraic fractions and solving the resulting linear equation. This problem is ideal for students in Grades 8-10 and covers key algebraic concepts.

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