The given expression is a quadratic equation in the form \(20z^2 + 13z - 15\). To solve it, we can factor it, if possible, or use the quadratic formula.

### Factoring
Let's try factoring first.

1. Multiply the coefficient of \(z^2\) (which is 20) by the constant term (which is -15):
   \[
   20 \times -15 = -300
   \]

2. We need to find two numbers that multiply to -300 and add to the coefficient of \(z\) (which is 13). After checking, we find that these numbers are 25 and -12.

3. Rewrite the middle term (13z) using the two numbers we found:
   \[
   20z^2 + 25z - 12z - 15
   \]

4. Factor by grouping:
   \[
   (20z^2 + 25z) - (12z + 15)
   \]
   Factor out the greatest common factor from each group:
   \[
   5z(4z + 5) - 3(4z + 5)
   \]
   Notice that \((4z + 5)\) is common, so we factor it out:
   \[
   (4z + 5)(5z - 3)
   \]

So, the factored form of the quadratic equation is:
\[
(4z + 5)(5z - 3) = 0
\]

### Finding the Solutions
Set each factor equal to zero to solve for \(z\):

1. \(4z + 5 = 0\)
   \[
   4z = -5 \implies z = -\frac{5}{4}
   \]

2. \(5z - 3 = 0\)
   \[
   5z = 3 \implies z = \frac{3}{5}
   \]

### Final Solutions
The solutions to the equation are:
\[
z = -\frac{5}{4} \quad \text{or} \quad z = \frac{3}{5}
\]

Would you like more details or have any questions about this solution?

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Here are some related questions:
1. How do you determine if a quadratic equation can be factored easily?
2. What is the quadratic formula and when should it be used?
3. How do you graph the solutions of a quadratic equation?
4. What are the applications of solving quadratic equations in real life?
5. How do you handle complex solutions in a quadratic equation?

**Tip:** Always check your factored solutions by multiplying the factors back to ensure they give the original quadratic expression.

Learn how to solve the quadratic equation 20z^2 + 13z - 15 with a detailed step-by-step solution. This problem covers key algebraic concepts, including factoring quadratic equations and using the quadratic formula. Suitable for students in Grades 10-12.

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