To simplify the inequality \( n(n - 6) \geq -4 \), let's go step by step.

### Step 1: Expand the left-hand side
First, distribute the \( n \) on the left-hand side:
\[
n(n - 6) = n^2 - 6n
\]
So the inequality becomes:
\[
n^2 - 6n \geq -4
\]

### Step 2: Move all terms to one side
To set the inequality to 0 on the right-hand side, add 4 to both sides:
\[
n^2 - 6n + 4 \geq 0
\]

### Step 3: Analyze the quadratic inequality
We now have a quadratic inequality:
\[
n^2 - 6n + 4 \geq 0
\]
To solve this, we can either factor (if possible) or use the quadratic formula. Since this doesn't factor easily, let's use the quadratic formula to find the critical points.

### Step 4: Apply the quadratic formula
The quadratic formula is:
\[
n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the equation \( n^2 - 6n + 4 = 0 \), the coefficients are:
- \( a = 1 \)
- \( b = -6 \)
- \( c = 4 \)

Now plug these values into the quadratic formula:
\[
n = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(4)}}{2(1)}
\]
\[
n = \frac{6 \pm \sqrt{36 - 16}}{2}
\]
\[
n = \frac{6 \pm \sqrt{20}}{2}
\]
\[
n = \frac{6 \pm 2\sqrt{5}}{2}
\]
\[
n = 3 \pm \sqrt{5}
\]

Thus, the critical points are:
\[
n = 3 + \sqrt{5} \quad \text{and} \quad n = 3 - \sqrt{5}
\]

### Step 5: Test intervals
Now, we test the intervals defined by these critical points to determine where \( n^2 - 6n + 4 \geq 0 \).

- For \( n < 3 - \sqrt{5} \), the quadratic expression is positive.
- For \( 3 - \sqrt{5} \leq n \leq 3 + \sqrt{5} \), the quadratic expression is negative.
- For \( n > 3 + \sqrt{5} \), the quadratic expression is positive.

Thus, the solution to the inequality is:
\[
n \leq 3 - \sqrt{5} \quad \text{or} \quad n \geq 3 + \sqrt{5}
\]

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

---

Would you like more details on any step or have any questions?

Here are 5 related questions for further practice:
1. Solve the inequality \( n(n + 3) \geq -5 \).
2. Simplify and solve \( n(n - 4) \geq 7 \).
3. Solve \( n^2 - 8n + 15 \leq 0 \).
4. Solve \( 2n^2 - 3n + 1 > 0 \).
5. Determine the solution set for \( n^2 + 2n - 8 \leq 0 \).

**Tip:** Always check for critical points when solving quadratic inequalities, as they help divide the number line into intervals.

Learn how to simplify the quadratic inequality n(n-6) ≥ -4 by expanding, solving, and applying the quadratic formula. This detailed step-by-step solution is perfect for students in Grades 9-11 looking to strengthen their understanding of algebraic inequalities and quadratic equations.

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