Use the functions f left parenthesis x right parenthesis equals x squared minus 3 x minus 18 and g left parenthesis x right parenthesis equals x squared plus 3 x minus 18 to answer parts​ (a) through​ (g).
​(a) Solve f left parenthesis x right parenthesis equals 0.
​(d) Solve f left parenthesis x right parenthesis greater than 0.
​(g) Solve f left parenthesis x right parenthesis greater than or equals 2.
​(b) Solve g left parenthesis x right parenthesis equals 0.
​(e) Solve g left parenthesis x right parenthesis less than or equals 0.
​(c) Solve f left parenthesis x right parenthesis equals g left parenthesis x right parenthesis.
​(f) Solve f left parenthesis x right parenthesis greater than g left parenthesis x right parenthesis.

Here are the solutions for each part:

**(a)** Solve \( f(x) = 0 \):  
The solutions are \( x = -3 \) and \( x = 6 \).

**(b)** Solve \( g(x) = 0 \):  
The solutions are \( x = -6 \) and \( x = 3 \).

**(c)** Solve \( f(x) = g(x) \):  
The solution is \( x = 0 \).

**(d)** Solve \( f(x) > 0 \):  
The solution is \( x \in (-\infty, -3) \cup (6, \infty) \).

**(e)** Solve \( g(x) \leq 0 \):  
The solution is \( x \in [-6, 3] \).

**(f)** Solve \( f(x) > g(x) \):  
The solution is \( x \in (-\infty, 0) \).

**(g)** Solve \( f(x) \geq 2 \):  
The solution is \( x \in (-\infty, \frac{3}{2} - \frac{\sqrt{89}}{2}] \cup [\frac{3}{2} + \frac{\sqrt{89}}{2}, \infty) \).

Would you like further details on any of these solutions?  

Here are five related questions for practice:
1. What are the intervals where \( g(x) > 0 \)?
2. How do the graphs of \( f(x) \) and \( g(x) \) compare?
3. What are the vertex points of both \( f(x) \) and \( g(x) \)?
4. Can you determine the axis of symmetry for each function?
5. How would \( f(x) \geq g(x) \) be solved?

**Tip:** When solving inequalities, always remember to test the critical points where the expressions are equal, as these can help determine the correct intervals.

Use the functions f(x) = x^2 - 3x - 18 and g(x) = x^2 + 3x - 18 to answer parts (a) through (g): (a) Solve f(x) = 0, (b) Solve g(x) = 0, (c) Solve f(x) = g(x), (d) Solve f(x) > 0, (e) Solve g(x) ≤ 0, (f) Solve f(x) > g(x), (g) Solve f(x) ≥ 2.

Learn how to solve quadratic equations and inequalities using the functions f(x) = x^2 - 3x - 18 and g(x) = x^2 + 3x - 18. This detailed solution covers solving equations, comparing functions, and handling inequalities. Suitable for students in Grades 9-12.

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