The function f is defined by f(x)=ax^(2)+bx+c, where a,b, and c are constants and 1

Question

The function  f is defined by  f(x)=ax^(2)+bx+c, where  a,b, and  c are constants and  1 < a < 4. The graph of  y=f(x) in the  xy-plane passes through points.  (11,0) and  (-2,0). If  a is an integer, what could be the value of  a+b ?

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Set up equations: Since the graph of y=f(x) passes through the points (11,0) and (-2,0), we can set up two equations using these points to find the relationship between a, b, and c. f(11) = a(11)^2 + b(11) + c = 0 f(-2) = a(-2)^2 + b(-2) + c = 0Substitute x-values: Let's substitute the x-values from the points into the equations to get: a(11)^2 + b(11) + c = 0 a(-2)^2 + b(-2) + c = 0Simplify equations: Now we simplify the equations: 121a + 11b + c = 0 4a - 2b + c = 0Express c in terms: We have two equations with three unknowns, which means we cannot find the exact values of a, b, and c without additional information. However, we can express c in terms of a and b using one of the equations. Let's use the second equation to express c: c = -4a + 2bSubstitute c into first equation: Substitute c from the fourth step into the first equation: 121a + 11b - 4a + 2b = 0 117a + 13b = 0Find possible values of a: We need to find the values of a and b such that a is an integer and 1 < a < 4. Since a must be an integer, let's list the possible values of a: 2, 3.Try a = 2: First, let's try a = 2 and solve for b using the equation 117a + 13b = 0: 117(2) + 13b = 0 234 + 13b = 0 13b = -234 b = -234 / 13 b = -18Try a = 3: Now let's check if a = 3 gives us an integer value for b: 117(3) + 13b = 0 351 + 13b = 0 13b = -351 b = -351 / 13 b = -27Calculate a + b: Both a = 2 and a = 3 give us integer values for b. Now we can calculate a + b for each case: For a = 2, b = -18, so a + b = 2 - 18 = -16. For a = 3, b = -27, so a + b = 3 - 27 = -24.Check valid values: Since the question asks for possible values of a + b, we have two possible answers: -16 and -24. However, we need to ensure that a + b is a single value, so we must check if both values of a are valid given the constraints of the problem.The constraints are 1 < a < 4 and a is an integer. Both a = 2 and a = 3 satisfy these constraints. Therefore, both -16 and -24 are valid answers for a + b.

The function $f$ is defined by $f(x)=a x^{2}+b x+c$ , where $a, b$ , and $c$ are constants and \( 1

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