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Range of Quadratic Expression

Let Px=ax2+bx...

Question

Let $P (x) = a x^{2} + b x + 8$ is a quadratic polynomial. If the minimum value of $P (x)$ is $6$ at $x = 2,$ then

A

a = \frac{1}{2}

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B

b = - \frac{1}{2}

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C

a = 2

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D

b = - 2

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Solution

The correct option is D $b = - 2$
$- \frac{b}{2 a} = 2$
$4 a = - b \dots (1)$
$P (2) = 4 a + 2 b + 8 = 6$
$4 a + 2 b = - 2 \dots (2)$

Using $(1)$ and $(2)$
$b = - 2, a = \frac{1}{2}$

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