If there no number between x^2 then z^2-12=4z can be easily simplified 

To solve the equation \( z^2 - 12 = 4z \) by simplifying, follow these steps:

### Step 1: Move all terms to one side of the equation
To set the equation to zero, subtract \( 4z \) from both sides:

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

### Step 2: Factor the quadratic equation
Now, we'll attempt to factor the quadratic equation \( z^2 - 4z - 12 = 0 \).

We need to find two numbers that multiply to \(-12\) (the constant term) and add to \(-4\) (the coefficient of \(z\)).

However, no such integers exist that satisfy this condition. Therefore, the quadratic equation cannot be factored easily with integers. 

### Step 3: Use the quadratic formula
Since factoring is not straightforward, we use the quadratic formula:

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

For the equation \( z^2 - 4z - 12 = 0 \), we have:
- \( a = 1 \) (coefficient of \( z^2 \))
- \( b = -4 \) (coefficient of \( z \))
- \( c = -12 \) (constant term)

Substituting these values into the quadratic formula:

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

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

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

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

### Step 4: Solve for \( z \)
Now, solve for the two possible values of \( z \):

1. \( z = \frac{4 + 8}{2} = \frac{12}{2} = 6 \)
2. \( z = \frac{4 - 8}{2} = \frac{-4}{2} = -2 \)

### Final Answer:
The solutions to the equation are:

\[
z = 6 \quad \text{or} \quad z = -2
\]

Would you like more details or have any questions? Here are some related questions:

1. How does the quadratic formula work in more complex equations?
2. What happens when the discriminant is negative?
3. How do you complete the square for a quadratic equation?
4. What is the significance of the vertex of a parabola in quadratic equations?
5. Can all quadratic equations be solved using the quadratic formula?

**Tip:** If factoring a quadratic equation seems challenging, the quadratic formula is a reliable method that always works.

Learn how to simplify and solve the quadratic equation z^2 - 12 = 4z step-by-step. Understand the quadratic formula and its application to solve for z. Suitable for high school students and anyone studying quadratic equations.

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