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Advanced Math / Nonlinear functions Difficulty: Hard

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

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Explanation

Choice A is correct. It's given that the graph of $y = f (x)$ in the xy-plane passes through the points $left parenthesis 7 comma 0 right parenthesis$ and $left parenthesis negative 3 comma 0 right parenthesis$ . It follows that when the value of $x$ is either $7$ or $negative 3$ , the value of $f (x)$ is $0$ . It's also given that the function $f$ is defined by $f (x) = a x^{2} + b x + c$ , where $a$ , $b$ , and $c$ are constants. It follows that the function $f$ is a quadratic function and, therefore, may be written in factored form as $f (x) = a (x - u) (x - v)$ , where the value of $f (x)$ is $0$ when $x$ is either $u$ or $v$ . Since the value of $f (x)$ is $0$ when the value of $x$ is either $7$ or $negative 3$ , and the value of $f (x)$ is $0$ when the value of $x$ is either $u$ or $v$ , it follows that $u$ and $v$ are equal to $7$ and $negative 3$ . Substituting $7$ for $u$ and $negative 3$ for $v$ in the equation $f (x) = a (x - u) (x - v)$ yields $f (x) = a (x - 7) (x - (- 3))$ , or $f (x) = a (x - 7) (x + 3)$ . Distributing the right-hand side of this equation yields $f (x) = a (x^{2} - 7 x + 3 x - 21)$ , or $f (x) = a x^{2} - 4 a x - 21 a$ . Since it's given that $f (x) = a x^{2} + b x + c$ , it follows that $b = - 4 a$ . Adding $a$ to each side of this equation yields $a + b = - 3 a$ . Since $a + b = - 3 a$ , if $a$ is an integer, the value of $a + b$ must be a multiple of $3$ . If $a$ is an integer greater than $1$ , it follows that $a greater than or equals 2$ . Therefore, $- 3 a \leq - 3 (2)$ . It follows that the value of $a + b$ is less than or equal to $minus 3 left parenthesis 2 right parenthesis$ , or $negative 6$ . Of the given choices, only $negative 6$ is a multiple of $3$ that's less than or equal to $negative 6$ .

Choice B is incorrect. This is the value of $a + b$ if $a$ is equal to, not greater than, $1$ .

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.