To solve the quadratic equation:

\[
12z^2 + 5z - 2 = 0
\]

we can use the **quadratic formula**:

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

Where \( a = 12 \), \( b = 5 \), and \( c = -2 \).

### Step-by-step solution:

1. **Substitute the values into the quadratic formula:**
   \[
   z = \frac{-5 \pm \sqrt{5^2 - 4(12)(-2)}}{2(12)}
   \]
   
2. **Simplify inside the square root:**
   \[
   z = \frac{-5 \pm \sqrt{25 + 96}}{24}
   \]
   \[
   z = \frac{-5 \pm \sqrt{121}}{24}
   \]
   
3. **Simplify the square root:**
   \[
   z = \frac{-5 \pm 11}{24}
   \]

4. **Find the two possible solutions:**

   - For \( z_1 \):
     \[
     z_1 = \frac{-5 + 11}{24} = \frac{6}{24} = \frac{1}{4}
     \]

   - For \( z_2 \):
     \[
     z_2 = \frac{-5 - 11}{24} = \frac{-16}{24} = \frac{-2}{3}
     \]

### Final solutions:
\[
z_1 = \frac{1}{4}, \quad z_2 = \frac{-2}{3}
\]

Would you like more details on any of the steps, or have any further questions?

### Related Questions:
1. What is the discriminant and how does it determine the nature of the roots?
2. Can we factorize this quadratic equation instead of using the formula?
3. What happens when the discriminant is negative?
4. How can we use the vertex form to analyze the graph of this equation?
5. What are the sum and product of the roots in this equation?

### Tip:
Always check if the quadratic equation can be factored easily before applying the quadratic formula to save time!

Learn how to solve the quadratic equation 12z^2 + 5z - 2 using the quadratic formula. This solution involves a step-by-step breakdown of the process, including discriminant analysis. Suitable for students in Grades 8-10.

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