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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 ?

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

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Conjugate root theorems

Full solution

Q. The function

f

is defined by

f(x)=a x^{2}+b x+c

, where

a, b

, and

c

are constants and

1<a<4

. The graph of

y=f(x)

in the

x y

-plane passes through points.

(11,0)

and

(-2,0)

. If

a

is an integer, what could be the value of

f(x)=a x^{2}+b x+c

0

?

Identify Roots of Quadratic: Since the graph of $f(x)$ passes through the points $(11,0)$ and $(-2,0)$ , we can say that $x=11$ and $x=-2$ are roots of the quadratic equation $ax^2+bx+c=0$ . We can use these roots to create two equations based on the fact that $f(11)=0$ and $f(-2)=0$ .
Substitute Roots into Equation: Substituting $x=11$ into the equation gives us $a(11)^2 + b(11) + c = 0$ . This simplifies to $121a + 11b + c = 0$ .
Create System of Equations: Substituting $x=-2$ into the equation gives us $a(-2)^2 + b(-2) + c = 0$ . This simplifies to $4a - 2b + c = 0$ .
Eliminate Variable $c$ : We now have a system of two equations with three unknowns: $\newline$ $1$ ) $121a + 11b + c = 0$ $\newline$ $2$ ) $4a - 2b + c = 0$ $\newline$ We need one more equation to solve for $a$ , $b$ , and $c$ . However, we are only asked to find the value of $a+b$ , not the individual values of $a$ , $b$ , and $c$ .
Solve for $b$ in terms of $a$ : We can eliminate $c$ by subtracting the second equation from the first equation: $\newline$ $(121a + 11b + c) - (4a - 2b + c) = 0 - 0$ $\newline$ This simplifies to $117a + 13b = 0$ .
Find $a+b$ : We can solve for $b$ in terms of $a$ by rearranging the equation: $\newline$ $b = -\frac{117a}{13}$ $\newline$ $b = -9a$
Determine Possible Values for $a$ : Now we can find $a+b$ by substituting $b = -9a$ into the expression: $\newline$ $a + b = a - 9a$ $\newline$ $a + b = -8a$
Calculate Possible Values for $a+b$ : Since $1 < a < 4$ and $a$ is an integer, the possible values for $a$ are $2$ and $3$ .
Final Answer: If $a = 2$ , then $a + b = -8(2) = -16$ . If $a = 3$ , then $a + b = -8(3) = -24$ .
Final Answer: If $a = 2$ , then $a + b = -8(2) = -16$ . If $a = 3$ , then $a + b = -8(3) = -24$ . Both $-16$ and $-24$ are possible values for $a+b$ given the constraints on $a$ . However, since the question asks for ＂what could be the value of $a+b$ ＂, we can provide any one of the possible values.

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