To solve the inequality:

\[
n(n - 6) \geq -4
\]

Let's start by expanding and rearranging the inequality:

\[
n^2 - 6n \geq -4
\]

Now, move \(-4\) to the left side to set the inequality to 0:

\[
n^2 - 6n + 4 \geq 0
\]

Next, we need to find the roots of the quadratic equation \(n^2 - 6n + 4 = 0\). We use the quadratic formula:

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

where \(a = 1\), \(b = -6\), and \(c = 4\). Substituting these values into the formula gives:

\[
n = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(4)}}{2(1)}
\]

Simplify the expression:

\[
n = \frac{6 \pm \sqrt{36 - 16}}{2}
\]

\[
n = \frac{6 \pm \sqrt{20}}{2}
\]

Since \(\sqrt{20} = 2\sqrt{5}\), we have:

\[
n = \frac{6 \pm 2\sqrt{5}}{2}
\]

\[
n = 3 \pm \sqrt{5}
\]

So, the roots of the quadratic are \(n = 3 + \sqrt{5}\) and \(n = 3 - \sqrt{5}\).

### Step 2: Determine the intervals

The roots divide the number line into three intervals:

1. \(n < 3 - \sqrt{5}\)
2. \(3 - \sqrt{5} \leq n \leq 3 + \sqrt{5}\)
3. \(n > 3 + \sqrt{5}\)

We need to determine where the quadratic expression \(n^2 - 6n + 4 \geq 0\) holds. The quadratic expression opens upwards (since the coefficient of \(n^2\) is positive), so it will be non-negative outside the interval between the roots.

### Step 3: Test the intervals

The inequality \(n^2 - 6n + 4 \geq 0\) is satisfied when:

1. \(n \leq 3 - \sqrt{5}\)
2. \(n \geq 3 + \sqrt{5}\)

### Conclusion

The solution to the inequality is:

\[
n \in (-\infty, 3 - \sqrt{5}] \cup [3 + \sqrt{5}, \infty)
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you solve quadratic inequalities with different coefficients?
2. What are the steps to graph quadratic functions and inequalities?
3. How do you use the quadratic formula for complex roots?
4. Can inequalities be solved using factorization or completing the square?
5. How does the discriminant of a quadratic equation relate to the number of real roots?

**Tip:** When solving quadratic inequalities, always check the direction of the parabola to determine where the expression is positive or negative.

Learn how to solve the quadratic inequality n(n-6) ≥ -4 with a detailed step-by-step solution. This problem involves key algebraic concepts such as quadratic inequalities, the quadratic formula, and understanding the properties of inequalities. Suitable for students in Grades 9-11.

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