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The question asks you to find the zeros of the function \( f(x) = -5x^2 + 4x - 1 \) by completing the square.

### Step-by-Step Solution
1. **Rewrite the Equation**: 
   To complete the square, start with the quadratic function:
   \[
   f(x) = -5x^2 + 4x - 1
   \]
   Factor out \(-5\) from the terms involving \(x\):
   \[
   f(x) = -5(x^2 - \frac{4}{5}x) - 1
   \]

2. **Complete the Square**:
   To complete the square inside the parentheses, take half of the coefficient of \(x\), square it, and add/subtract it:
   - The coefficient of \(x\) in \(x^2 - \frac{4}{5}x\) is \(-\frac{4}{5}\).
   - Half of \(-\frac{4}{5}\) is \(-\frac{2}{5}\).
   - Squaring it gives \(\left(\frac{2}{5}\right)^2 = \frac{4}{25}\).
   
   Add and subtract \(\frac{4}{25}\) inside the parentheses:
   \[
   f(x) = -5\left(x^2 - \frac{4}{5}x + \frac{4}{25} - \frac{4}{25}\right) - 1
   \]
   \[
   f(x) = -5\left(\left(x - \frac{2}{5}\right)^2 - \frac{4}{25}\right) - 1
   \]

3. **Simplify the Expression**:
   Now distribute \(-5\):
   \[
   f(x) = -5\left(x - \frac{2}{5}\right)^2 + 1 - 1
   \]
   \[
   f(x) = -5\left(x - \frac{2}{5}\right)^2 + \frac{5}{25} - 1
   \]
   \[
   f(x) = -5\left(x - \frac{2}{5}\right)^2 + \frac{9}{25}
   \]

4. **Set \( f(x) = 0 \) to find the Zeros**:
   Set the equation to zero and solve for \(x\):
   \[
   0 = -5\left(x - \frac{2}{5}\right)^2 + \frac{5}{25} - 1
   \]
   Rearrange terms to isolate the squared term:
   \[
   \left(x - \frac{2}{5}\right)^2 = \frac{9}{25}
   \]
   Take the square root of both sides:
   \[
   x - \frac{2}{5} = \pm \frac{3}{5}
   \]
   So,
   \[
   x = \frac{2}{5} \pm \frac{3}{5}
   \]
   
5. **Find the Two Solutions**:
   Calculate each solution:
   - \( x = \frac{2}{5} + \frac{3}{5} = 1\)
   - \( x = \frac{2}{5} - \frac{3}{5} = -\frac{1}{5}\)

### Final Answer
The zeros of the function are:
\[
x = 1 \quad \text{and} \quad x = -\frac{1}{5}
\]

Would you like further clarification on this process, or do you have any questions?

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Here are some follow-up questions to deepen understanding:
1. How do you determine whether to complete the square or use another method like factoring?
2. What is the importance of completing the square for finding the vertex of a parabola?
3. How would you solve \( f(x) = 0 \) if the coefficient of \(x^2\) were positive instead?
4. What other methods can be used to find the zeros of a quadratic function?
5. How does completing the square relate to the quadratic formula?

**Tip:** Completing the square is a useful skill not only for solving quadratic equations but also for rewriting functions in a way that reveals their vertex form, which can be handy in graphing and analysis.

Find the zeros of the function f(x) = -5x^2 + 4x - 1 by completing the square.

Learn how to find the zeros of the function f(x) = -5x^2 + 4x - 1 by completing the square. This step-by-step solution covers key algebraic concepts, including quadratic equations and the completing the square technique, perfect for students in Grades 9-10.

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