solve it---26 The function f 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 ? A) -6 B) -3 C) 4 D) 5

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Let's solve this quadratic function problem. We're given that f of x equals a x squared plus b x plus c, where the graph passes through the points (7, 0) and (-3, 0). This means that x equals 7 and x equals negative 3 are the roots of the equation. Since a is an integer greater than 1, we need to find the possible value of a plus b from the given options. Since the graph passes through the points (7, 0) and (-3, 0), these are the roots of the quadratic function. We can use these roots to write the function in factored form. For a quadratic function with roots r1 and r2, we can write f(x) = a times (x minus r1) times (x minus r2). Substituting our roots, we get f(x) = a times (x minus 7) times (x minus negative 3), which simplifies to a times (x minus 7) times (x plus 3). Now, let's expand the factored form to get the standard form of the quadratic function. Starting with f(x) = a times (x minus 7) times (x plus 3), we multiply the terms inside the parentheses. This gives us a times (x squared plus 3x minus 7x minus 21), which simplifies to a times (x squared minus 4x minus 21). Distributing the coefficient a, we get f(x) = a x squared minus 4a x minus 21a. Comparing this with the standard form f(x) = a x squared plus b x plus c, we can identify that b equals negative 4a and c equals negative 21a. Now we can find the value of a plus b. We know that b equals negative 4a, so a plus b equals a plus negative 4a, which simplifies to negative 3a. Since we're told that a is an integer greater than 1, let's try some values. If a equals 2, then a plus b equals negative 3 times 2, which is negative 6. If a equals 3, then a plus b equals negative 9. If a equals 4, then a plus b equals negative 12. Looking at the given options, we see that negative 6 matches option A. Therefore, a plus b could equal negative 6. Let's summarize what we've learned. We started with a quadratic function f(x) = a x squared plus b x plus c that passes through the points (7, 0) and (-3, 0). Using these roots, we wrote the function in factored form as f(x) = a times (x minus 7) times (x plus 3). Expanding this expression, we got f(x) = a x squared minus 4a x minus 21a. By comparing coefficients with the standard form, we found that b equals negative 4a. Therefore, a plus b equals negative 3a. Since a must be an integer greater than 1, we tried a equals 2, which gave us a plus b equals negative 6. This matches option A, so our answer is negative 6.