Question

Let $f(x)$ be a quadratic polynomial in x such that $f(x)≥0$ for all real numbers x.If $f(2)=0$ and $f(4)=6$,then $f(−2)$ is equal to

• A. 12
• B. 36
• C. 24
• D. 6

Answer 1

Answer: (C)

24

Answer 2

The correct answer is C:24
Given that the quadratic expression is always greater than or equal to 0 for any real number x,it indicates that its graph forms a U-shaped curve opening upwards.
Given the points (2,0) and (4, 6) on the curve, it means the lowest point (vertex) of the curve is at (2,0).
The equation of a quadratic expression can be written as $y=a(x-h)^2+k$,where (h, k) represents the vertex.
Using the vertex coordinates (2, 0),the quadratic expression takes the form $y=a(x - 2)^2$.
Using the fact that y=6 when x=4, we can calculate the value of a:
$6=a(4 - 2)^2$
6=4a
$a=\frac{6}{4}=\frac{3}{2}$
So, the quadratic expression becomes $y = \frac{3}{2} \times (x - 2)^2$.
Now, let's find the value of the expression when x = -2:
$y = \frac{3}{2} \times (-2 - 2)^2$
$y = \frac{3}{2} \times (-4)^2$
$y = \frac{3}{2} \times 16$
y=24
Therefore, when x=-2, the value of the expression is 24.
The correct answer is option c. 24