https://file.aiquickdraw.com/mathbot/file/1730675693014b2uz6fiz.jpg

To solve the quadratic equation given in the image:

\[
-5z^2 - 4z + 5 = 0
\]

we can use the quadratic formula:

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

For this equation, \( a = -5 \), \( b = -4 \), and \( c = 5 \).

### Step 1: Calculate the Discriminant
The discriminant is \( b^2 - 4ac \):

\[
(-4)^2 - 4(-5)(5) = 16 + 100 = 116
\]

Since the discriminant is positive, there are two real solutions.

### Step 2: Substitute into the Quadratic Formula
\[
z = \frac{-(-4) \pm \sqrt{116}}{2(-5)} = \frac{4 \pm \sqrt{116}}{-10}
\]

### Step 3: Simplify
The square root of 116 can be simplified as \( \sqrt{116} = 2\sqrt{29} \).

Thus,

\[
z = \frac{4 \pm 2\sqrt{29}}{-10} = \frac{-2 \pm \sqrt{29}}{5}
\]

### Solution
The solutions are:

\[
z = \frac{-2 + \sqrt{29}}{5} \quad \text{and} \quad z = \frac{-2 - \sqrt{29}}{5}
\]

So, the correct answer is:
**A. The solution(s) is(are) \( z = \frac{-2 + \sqrt{29}}{5}, \frac{-2 - \sqrt{29}}{5} \)**.

Would you like more details on any steps, or do you have questions?

---

Here are five related questions that might expand your understanding:

1. What is the importance of the discriminant in determining the type of solutions?
2. How would the solution change if the discriminant was zero?
3. Can you solve the equation by completing the square instead of using the quadratic formula?
4. How does changing the sign of \( a \) affect the parabola of a quadratic equation?
5. Can a quadratic equation have only one real solution? If so, under what condition?

**Tip:** Always check the discriminant first to determine if you’ll have real or complex solutions.

Solve the quadratic equation: -5z^2 - 4z + 5 = 0

Learn how to solve the quadratic equation -5z^2 - 4z + 5 = 0 using the quadratic formula. This solution includes step-by-step guidance on calculating the discriminant and simplifying the square root. Suitable for high school students in Grades 9-12.

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