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The given task is to solve the inequality:

\[
-4x^2 + 5x - 1 \geq 0
\]

### Step 1: Solve the quadratic equation

First, find the roots of the quadratic equation:

\[
-4x^2 + 5x - 1 = 0
\]

We use the quadratic formula:

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

Here, \( a = -4 \), \( b = 5 \), and \( c = -1 \). Substituting these values into the formula:

\[
x = \frac{-5 \pm \sqrt{5^2 - 4(-4)(-1)}}{2(-4)}
\]
\[
x = \frac{-5 \pm \sqrt{25 - 16}}{-8}
\]
\[
x = \frac{-5 \pm \sqrt{9}}{-8}
\]
\[
x = \frac{-5 \pm 3}{-8}
\]

Thus, the two solutions are:

\[
x_1 = \frac{-5 + 3}{-8} = \frac{-2}{-8} = \frac{1}{4}
\]
\[
x_2 = \frac{-5 - 3}{-8} = \frac{-8}{-8} = 1
\]

### Step 2: Determine the sign of the quadratic expression

Now, we analyze the sign of the expression \( -4x^2 + 5x - 1 \) around the roots \( x_1 = \frac{1}{4} \) and \( x_2 = 1 \).

- The quadratic function opens downward (since the coefficient of \( x^2 \) is negative).
- The roots divide the real number line into three intervals: \( (-\infty, \frac{1}{4}) \), \( (\frac{1}{4}, 1) \), and \( (1, +\infty) \).

We check the sign of the quadratic expression in each of these intervals:

1. **For \( x < \frac{1}{4} \)**, choose \( x = 0 \):

   \[
   -4(0)^2 + 5(0) - 1 = -1
   \]

   The expression is negative.

2. **For \( \frac{1}{4} < x < 1 \)**, choose \( x = \frac{1}{2} \):

   \[
   -4\left(\frac{1}{2}\right)^2 + 5\left(\frac{1}{2}\right) - 1 = -4\left(\frac{1}{4}\right) + \frac{5}{2} - 1 = -1 + 2.5 - 1 = 0.5
   \]

   The expression is positive.

3. **For \( x > 1 \)**, choose \( x = 2 \):

   \[
   -4(2)^2 + 5(2) - 1 = -16 + 10 - 1 = -7
   \]

   The expression is negative.

### Step 3: Write the solution

We want the quadratic expression to be non-negative:

\[
-4x^2 + 5x - 1 \geq 0
\]

From the sign analysis, the expression is non-negative in the interval \( \left[\frac{1}{4}, 1\right] \).

Thus, the solution to the inequality is:

\[
x \in \left[\frac{1}{4}, 1\right]
\]

### Step 4: Check the answer choices

Looking at the provided answer choices, the correct one is:

**(б)** \( \left[\frac{1}{4}, 1\right] \).

---

Would you like any further details or clarifications? Here are 5 questions that expand on this solution:

1. How does the quadratic formula help find the roots of a quadratic equation?
2. Why is it important to check the sign of the quadratic expression in different intervals?
3. How do you determine the behavior of a quadratic function based on the coefficient of \(x^2\)?
4. Can you explain why the solution is a closed interval \( [\frac{1}{4}, 1] \) instead of an open one?
5. What would happen if the inequality was strict, i.e., \( -4x^2 + 5x - 1 > 0 \)?

**Tip:** Always verify your solution by substituting values from the interval back into the original inequality.

Solve the inequality: -4x^2 + 5x - 1 ≥ 0.

Learn how to solve the quadratic inequality -4x^2 + 5x - 1 ≥ 0 with a step-by-step solution. This problem involves using the quadratic formula and analyzing intervals to determine where the expression is non-negative. Suitable for students in Grades 9-12.

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