The statement $$ \left(n+2\right)^2 $$$$ <4n^2 $$ true or false?


Let's analyze the inequality $ \left(n + 2\right)^2 < 4n^2 $ to determine whether it's true or false.

### Step 1: Expand the left-hand side
Expand $ \left(n + 2\right)^2 $:

\[
\left(n + 2\right)^2 = n^2 + 4n + 4
\]

### Step 2: Rewrite the inequality
The inequality becomes:

\[
n^2 + 4n + 4 < 4n^2
\]

### Step 3: Move all terms to one side of the inequality
Subtract $ n^2 + 4n + 4 $ from both sides:

\[
0 < 4n^2 - n^2 - 4n - 4
\]

Simplifying this:

\[
0 < 3n^2 - 4n - 4
\]

### Step 4: Analyze the quadratic inequality
We now have the quadratic inequality:

\[
3n^2 - 4n - 4 > 0
\]

To determine where this inequality is true, let's find the roots of the quadratic equation by setting it equal to zero:

\[
3n^2 - 4n - 4 = 0
\]

Use the quadratic formula:

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

where $ a = 3 $, $ b = -4 $, and $ c = -4 $.

\[
n = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(3)(-4)}}{2(3)}
\]
\[
n = \frac{4 \pm \sqrt{16 + 48}}{6}
\]
\[
n = \frac{4 \pm \sqrt{64}}{6}
\]
\[
n = \frac{4 \pm 8}{6}
\]

This gives us two roots:

\[
n = \frac{4 + 8}{6} = 2 \quad \text{and} \quad n = \frac{4 - 8}{6} = -\frac{2}{3}
\]

### Step 5: Test the intervals
The roots divide the number line into three intervals: $ (-\infty, -\frac{2}{3}) $, $ \left(-\frac{2}{3}, 2\right) $, and $ (2, \infty) $.

- For $ n > 2 $, $ 3n^2 - 4n - 4 > 0 $, so the inequality holds.
- For $ n < -\frac{2}{3} $, $ 3n^2 - 4n - 4 > 0 $, so the inequality holds.
- For $ -\frac{2}{3} < n < 2 $, $ 3n^2 - 4n - 4 < 0 $, so the inequality does **not** hold.

### Conclusion:
The inequality $ \left(n + 2\right)^2 < 4n^2 $ is **false** for $ n $ in the interval $ -\frac{2}{3} < n < 2 $, and **true** for $ n $ outside this interval.

Would you like more details or have any questions?

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Here are some related questions:

1. What is the significance of the roots in solving inequalities?
2. How do we determine where a quadratic inequality is true or false?
3. How do we use the quadratic formula to find the roots of a quadratic equation?
4. What are some other methods to solve quadratic inequalities?
5. How can we graphically interpret the solution to the inequality?
6. What happens if we change the inequality sign to $ \leq $ or $ \geq $?
7. How does the interval test work for quadratic inequalities?
8. Can we solve inequalities involving cubic equations using similar methods?

**Tip:** When solving quadratic inequalities, finding the roots is key to determining the intervals where the inequality holds.

Explore the detailed analysis of the inequality (n + 2)^2 < 4n^2, including step-by-step solution techniques. This problem covers quadratic inequalities, involving the quadratic formula. Suitable for high school students in Grades 10-12.

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