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Let's solve the equation \( z^2 - 12 = 4z \) by factoring.

### Step 1: Rearrange the equation
First, move all the terms to one side of the equation to set it equal to zero:

\[
z^2 - 4z - 12 = 0
\]

### Step 2: Factor the quadratic equation
We need to factor the quadratic expression \( z^2 - 4z - 12 \). To do this, look for two numbers that multiply to \(-12\) (the constant term) and add to \(-4\) (the coefficient of \( z \)).

However, since it may not be easy to factor by simple inspection, you might need to use the quadratic formula instead:

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

For the equation \( z^2 - 4z - 12 = 0 \):

- \(a = 1\)
- \(b = -4\)
- \(c = -12\)

### Step 3: Apply the quadratic formula
Plugging the values into the formula:

\[
z = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-12)}}{2(1)}
\]

\[
z = \frac{4 \pm \sqrt{16 + 48}}{2}
\]

\[
z = \frac{4 \pm \sqrt{64}}{2}
\]

\[
z = \frac{4 \pm 8}{2}
\]

This gives us two possible solutions:

\[
z = \frac{4 + 8}{2} = \frac{12}{2} = 6
\]

\[
z = \frac{4 - 8}{2} = \frac{-4}{2} = -2
\]

### Final Answer:
The solutions to the equation \( z^2 - 12 = 4z \) are \( z = 6 \) and \( z = -2 \).

Would you like more details on any part of this solution or have any further questions?

Here are some related questions to expand your understanding:
1. How can you verify that the solutions \( z = 6 \) and \( z = -2 \) are correct?
2. What does the quadratic formula tell us about the nature of the roots?
3. Can you factor the quadratic expression \( z^2 - 4z - 12 \) by completing the square instead?
4. What are some other methods to solve a quadratic equation besides factoring and using the quadratic formula?
5. How can you graphically interpret the solutions of the quadratic equation?

**Tip:** Always check if a quadratic expression can be factored easily before applying the quadratic formula—it might save you some time!

Learn how to solve the quadratic equation z^2 - 12 = 4z with a detailed step-by-step solution. This problem covers key algebraic concepts, including factoring and the quadratic formula. Suitable for students in Grades 9-12.

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