https://file.aiquickdraw.com/mathbot/file/1727643363094o69nhf72.jpg

The problem in the image is asking about a quadratic function \( f(x) = ax^2 + bx + c \) where \( a \), \( b \), and \( c \) are constants. The graph passes through the points \( (7, 0) \) and \( (-3, 0) \). We are tasked with finding the value of \( a + b \), given that \( a \) is an integer greater than 1.

### Steps:
1. **Roots of the Quadratic**: Since \( (7, 0) \) and \( (-3, 0) \) are points on the graph, they are the roots of the quadratic equation. Therefore, the quadratic can be written in factored form:
   \[
   f(x) = a(x - 7)(x + 3)
   \]
   Expanding this expression gives:
   \[
   f(x) = a(x^2 + 3x - 7x - 21) = a(x^2 - 4x - 21)
   \]
   So, the quadratic function becomes:
   \[
   f(x) = ax^2 - 4ax - 21a
   \]

2. **Identifying \( a \) and \( b \)**:
   From this expanded form, we can identify that:
   - \( b = -4a \)
   - We are asked to find \( a + b \), which is:
     \[
     a + b = a - 4a = -3a
     \]

3. **Finding \( a + b \)**: Since \( a \) is an integer greater than 1, the smallest possible value of \( a \) is 2. Let's calculate \( a + b \) for \( a = 2 \):
   \[
   a + b = -3a = -3(2) = -6
   \]

Thus, the value of \( a + b \) is \( \boxed{-6} \).

Would you like more details or have any further questions?

Here are 5 related questions to consider:
1. How do you factor a quadratic equation given its roots?
2. What are other methods to find the vertex of a quadratic function?
3. How does the discriminant of a quadratic equation relate to its graph?
4. Can the quadratic function have more than two real roots?
5. How does the value of \( a \) in \( ax^2 + bx + c \) affect the shape of the parabola?

**Tip**: When dealing with quadratic functions, factoring is often the easiest method to identify roots and rewrite the equation in standard form.

The function f(x) is defined by f(x) = ax^2 + bx + c, where a, b, and c are constants. The graph of y = f(x) in the xy-plane passes through the points (7, 0) and (-3, 0). If a is an integer greater than 1, which of the following could be the value of a + b?

Learn how to solve a quadratic function in standard form, given two points where the function intersects the x-axis. Understand how to use factoring and the root theorem to find the values of constants a and b, given that a is an integer greater than 1. Suitable for Grades 10-12.

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