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Question 6
Let $$f(x)$$ be a quadratic polynomial in $$x$$ such that $$f(x) \geq 0$$ for all real numbers $$x$$. If f(2) = 0 and f( 4) = 6, then f(-2) is equal to
Solution
$$f(x) \geq 0$$for all real numbers $$x$$, so D<=0
Since f(2)=0 therefore x=2 is a root of f(x)
Since the discriminant of f(x) is less than equal to 0 and 2 is a root so we can conclude that D=0
Therefore f(x) = $$a\left(x-2\right)^2$$
f(4)=6
or, 6 = $$a\left(x-2\right)^2$$
a= 3/2
$$f\left(-2\right)=\ -\frac{3}{2}\left(-4\right)^2=24$$
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