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

To solve the polynomial equation $ p(x) = ax^2 + 7x + 2 $ when $ x = 3 $, we'll substitute $ x = 3 $ into the equation and simplify.

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

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

Simplify:
\[ p(3) = 9a + 21 + 2 \]

Combine like terms:
\[ p(3) = 9a + 23 \]

The solution to the polynomial equation when $ x = 3 $ is:
\[ p(3) = 9a + 23 \]

Would you like to further explore this equation, such as finding the value of $ p(x) $ for a specific value of $ a $ or any other details?

Here are 5 questions related to this topic:

1. What happens to $ p(x) $ if the coefficient $ a $ is 0?
2. How would the solution change if we were solving for a different value of $ x $?
3. Can you find the roots of $ p(x) = 0 $ for a given $ a $?
4. How does the shape of the graph of $ p(x) $ change with different values of $ a $?
5. What is the vertex of the parabola represented by $ p(x) = ax^2 + 7x + 2 $?

**Tip:** When substituting values into polynomials, ensure you simplify step by step to avoid mistakes.

Learn how to solve the polynomial equation ax^2 + 7x + 2 when x = 3 with a detailed step-by-step solution. This problem covers key algebraic concepts, including polynomial equations and quadratic formulas. Suitable for students in Grades 7-9.

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