We are given the inequality:

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

Let's solve this step by step.

### Step 1: Expand the equation
First, expand the left-hand side of the inequality:

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

### Step 2: Move all terms to one side
Move \(-4\) to the other side to set the inequality to 0:

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

### Step 3: Solve the quadratic inequality
We now need to solve the quadratic inequality \(n^2 - 6n + 4 \geq 0\). First, find the roots of the equation by using the quadratic formula:

\[
n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the quadratic \(n^2 - 6n + 4 = 0\), the coefficients are \(a = 1\), \(b = -6\), and \(c = 4\). Substituting these into the quadratic formula:

\[
n = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(4)}}{2(1)}
\]
\[
n = \frac{6 \pm \sqrt{36 - 16}}{2}
\]
\[
n = \frac{6 \pm \sqrt{20}}{2}
\]
\[
n = \frac{6 \pm 2\sqrt{5}}{2}
\]
\[
n = 3 \pm \sqrt{5}
\]

So, the two roots are:
\[
n_1 = 3 + \sqrt{5}, \quad n_2 = 3 - \sqrt{5}
\]

### Step 4: Determine the intervals
The quadratic inequality \(n^2 - 6n + 4 \geq 0\) will be true outside the roots. Thus, the solution will be in the intervals:
\[
(-\infty, 3 - \sqrt{5}] \cup [3 + \sqrt{5}, \infty)
\]

This is because quadratic expressions are positive outside their roots and negative between them.

### Final Answer:
The solution to the inequality is:
\[
n \in (-\infty, 3 - \sqrt{5}] \cup [3 + \sqrt{5}, \infty)
\]

---

Would you like further explanation on any of the steps?

Here are 5 related questions you might find interesting:
1. How do you graph quadratic inequalities on a number line?
2. Can you solve a quadratic inequality using a different method?
3. What happens if the inequality is strict, i.e., \(n(n - 6) > -4\)?
4. How does the discriminant in the quadratic formula help in determining the solution?
5. How do you solve an inequality involving absolute values?

**Tip:** Always check the sign of the quadratic expression on intervals determined by its roots to find the correct solution!

Learn how to solve the quadratic inequality n(n - 6) ≥ -4 with a step-by-step solution. This detailed guide covers quadratic equations, expanding expressions, and solving inequalities using the quadratic formula. Suitable for students in Grades 9-11.

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