$$ p $$($$ x $$) = $$ ax $$$$ \placeholder{}^2 $$+$$ 7x+12\text{ when x}=3\text{ solve the polynomial equation} $$

Let's solve the polynomial equation $ p(x) = ax^2 + 7x + 12 $ when $ x = 3 $.

Given:
\[ p(x) = ax^2 + 7x + 12 \]

Substitute $ x = 3 $ into the equation:
\[ p(3) = a(3)^2 + 7(3) + 12 \]

Now, simplify:
\[ p(3) = 9a + 21 + 12 \]

Combine like terms:
\[ p(3) = 9a + 33 \]

So, the solution to the polynomial equation when $ x = 3 $ is:
\[ p(3) = 9a + 33 \]

Would you like to calculate $ p(3) $ for a specific value of $ a $ or have any other questions?

Here are 5 questions related to this problem:

1. What is the value of $ p(3) $ if $ a = 2 $?
2. How would the equation change if the constant term was different?
3. Can you find the value of $ x $ that makes $ p(x) = 0 $ for a given $ a $?
4. What would $ p(x) $ look like if $ a $ were negative?
5. How does the coefficient $ a $ affect the steepness of the parabola?

**Tip:** Understanding how each term in a polynomial affects the graph helps in predicting the behavior of the equation when different values are substituted.

Learn how to solve the polynomial equation ax^2 + 7x + 12 when x = 3 with a detailed step-by-step solution. This problem involves polynomial functions and quadratic equations, specifically addressing the evaluation of polynomial expressions at specific points. Suitable for high school students in Grades 9-12.

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