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Range of Quadratic Expression

The graph of ...

Question

The graph of $f (x) = a x^{2} + b x + c$ is shown below, such that $b^{2} - 4 a c = - 4.$ If the length of segment $A B$ and $A C$ are $1$ and $4$ respectively, then the value of $(a + b + c)$ is equal to

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Solution

$f (x) = a x^{2} + b x + c$
$D = b^{2} - 4 a c = - 4 . . . (1)$
Least value of $f (x) = - \frac{D}{4 a} = ¯ ¯¯¯¯¯¯ ¯ A B$
$\Rightarrow - \frac{D}{4 a} = 1$
$\Rightarrow - (- 4) = - 4 a [From (1)]$
$\Rightarrow a = 1$

Also $- \frac{b}{2 a} = - 4 \Rightarrow b = 8$
And $b^{2} - 4 a c = - 4$
$\Rightarrow 64 - 4 c = - 4$
$\Rightarrow c = 17$
Hence, $a + b + c = 1 + 8 + 17 = 26$

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