https://file.aiquickdraw.com/mathbot/file/17381351705071gtexqm6.jpg

Let's solve problem **27**, where the given functions are:

\[
f(x) = x^2 - 4
\]

\[
g(x) = -x^2 + 4
\]

We will solve the following conditions:

### **(a) Solve \( f(x) = 0 \)**
Set \( f(x) = 0 \):

\[
x^2 - 4 = 0
\]

Factorize:

\[
(x - 2)(x + 2) = 0
\]

So, the solutions are:

\[
x = \pm 2
\]

### **(b) Solve \( g(x) = 0 \)**
Set \( g(x) = 0 \):

\[
- x^2 + 4 = 0
\]

Rearrange:

\[
x^2 = 4
\]

Take the square root:

\[
x = \pm 2
\]

### **(c) Solve \( f(x) = g(x) \)**
Set \( f(x) = g(x) \):

\[
x^2 - 4 = -x^2 + 4
\]

Rearrange:

\[
x^2 + x^2 = 4 + 4
\]

\[
2x^2 = 8
\]

\[
x^2 = 4
\]

\[
x = \pm 2
\]

### **(d) Solve \( f(x) > 0 \)**
Solve:

\[
x^2 - 4 > 0
\]

Factor:

\[
(x - 2)(x + 2) > 0
\]

Find sign changes using test points:
- For \( x < -2 \), both factors are negative, so product is **positive**.
- For \( -2 < x < 2 \), one factor is negative and one is positive, so product is **negative**.
- For \( x > 2 \), both factors are positive, so product is **positive**.

Thus, the solution is:

\[
x \in (-\infty, -2) \cup (2, \infty)
\]

### **(e) Solve \( g(x) \leq 0 \)**
Solve:

\[
- x^2 + 4 \leq 0
\]

Rearrange:

\[
x^2 \geq 4
\]

Take square root:

\[
|x| \geq 2
\]

Thus, the solution is:

\[
x \in (-\infty, -2] \cup [2, \infty)
\]

### **(f) Solve \( f(x) > g(x) \)**
Solve:

\[
x^2 - 4 > -x^2 + 4
\]

Rearrange:

\[
x^2 + x^2 > 8
\]

\[
2x^2 > 8
\]

\[
x^2 > 4
\]

\[
|x| > 2
\]

Thus, the solution is:

\[
x \in (-\infty, -2) \cup (2, \infty)
\]

### **(g) Solve \( f(x) \geq 1 \)**
Solve:

\[
x^2 - 4 \geq 1
\]

\[
x^2 \geq 5
\]

\[
|x| \geq \sqrt{5}
\]

Thus, the solution is:

\[
x \in (-\infty, -\sqrt{5}] \cup [\sqrt{5}, \infty)
\]

---

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### **Related Questions**
1. How do you determine intervals for quadratic inequalities?
2. What is the graphical interpretation of \( f(x) = g(x) \)?
3. How do you test intervals for inequality solutions?
4. Why does the function \( f(x) = x^2 - 4 \) have the same roots as \( g(x) = -x^2 + 4 \)?
5. How would the solutions change if we used different functions?

### **Tip**  
For quadratic inequalities, always test points in each interval determined by the critical values!

Learn how to solve quadratic equations and inequalities for f(x) = x^2 - 4 and g(x) = -x^2 + 4 in Problem 27. This solution covers solving f(x) = 0, g(x) = 0, f(x) = g(x), and inequalities such as f(x) > 0 and f(x) > g(x). Step-by-step explanations included.

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