To solve the inequality \( 3x^2 - 11x - 4 \leq 0 \), we'll first find the roots of the quadratic equation \( 3x^2 - 11x - 4 = 0 \) and then determine the intervals where the quadratic expression is less than or equal to zero.

### Step 1: Find the roots of the quadratic equation
We'll use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the equation \( 3x^2 - 11x - 4 = 0 \), the coefficients are:
- \( a = 3 \)
- \( b = -11 \)
- \( c = -4 \)

Substituting these values into the quadratic formula:
\[
x = \frac{-(-11) \pm \sqrt{(-11)^2 - 4(3)(-4)}}{2(3)}
\]
\[
x = \frac{11 \pm \sqrt{121 + 48}}{6}
\]
\[
x = \frac{11 \pm \sqrt{169}}{6}
\]
\[
x = \frac{11 \pm 13}{6}
\]
So, the roots are:
\[
x_1 = \frac{11 + 13}{6} = \frac{24}{6} = 4
\]
\[
x_2 = \frac{11 - 13}{6} = \frac{-2}{6} = -\frac{1}{3}
\]

### Step 2: Determine the intervals
The roots split the number line into three intervals:
1. \( (-\infty, -\frac{1}{3}) \)
2. \( \left(-\frac{1}{3}, 4\right) \)
3. \( (4, \infty) \)

We'll test the sign of the quadratic expression \( 3x^2 - 11x - 4 \) in each interval.

- **For \( x \in (-\infty, -\frac{1}{3}) \)**:
  Choose \( x = -1 \):
  \[
  3(-1)^2 - 11(-1) - 4 = 3 + 11 - 4 = 10 \quad (\text{positive})
  \]
- **For \( x \in \left(-\frac{1}{3}, 4\right) \)**:
  Choose \( x = 0 \):
  \[
  3(0)^2 - 11(0) - 4 = -4 \quad (\text{negative})
  \]
- **For \( x \in (4, \infty) \)**:
  Choose \( x = 5 \):
  \[
  3(5)^2 - 11(5) - 4 = 75 - 55 - 4 = 16 \quad (\text{positive})
  \]

### Step 3: Write the solution
The quadratic expression \( 3x^2 - 11x - 4 \) is less than or equal to zero in the interval where the expression is negative, including the points where it equals zero:
\[
-\frac{1}{3} \leq x \leq 4
\]

Thus, the solution to the inequality \( 3x^2 - 11x - 4 \leq 0 \) is:
\[
x \in \left[-\frac{1}{3}, 4\right]
\]

Would you like more details on any part of this solution?

Here are some related questions to expand your understanding:

1. How do you solve a quadratic inequality when the discriminant is zero?
2. What does it mean when the quadratic inequality has no real roots?
3. How can you graphically interpret the solution to \( 3x^2 - 11x - 4 \leq 0 \)?
4. What is the difference between solving \( 3x^2 - 11x - 4 \leq 0 \) and \( 3x^2 - 11x - 4 < 0 \)?
5. How can you solve a quadratic inequality using a sign chart?

**Tip:** Always double-check your solutions by substituting them back into the original inequality to ensure they satisfy the condition.

Learn how to solve the quadratic inequality 3x^2 - 11x - 4 <= 0 with a detailed step-by-step solution. This problem covers quadratic equations and inequalities, including the quadratic formula. Suitable for high school students in Grades 9-12.

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