To determine which of the given functions is quadratic, we must first understand the characteristics of quadratic functions. A quadratic function is a polynomial function of degree 2, which means the highest power of the variable [tex]\( x \)[/tex] is [tex]\( x^2 \)[/tex].
Let's evaluate each choice step-by-step:
### Choice A: [tex]\( f(x)=3(x-4)(x+3) \)[/tex]
Let's expand this expression:
[tex]\[
f(x) = 3(x-4)(x+3)
\][/tex]
First, apply the distributive property (FOIL method):
[tex]\[
(x-4)(x+3) = x \cdot x + x \cdot 3 - 4 \cdot x - 4 \cdot 3 = x^2 + 3x - 4x - 12 = x^2 - x - 12
\][/tex]
Then multiply everything by 3:
[tex]\[
f(x) = 3(x^2 - x - 12) = 3x^2 - 3x - 36
\][/tex]
The highest power of [tex]\( x \)[/tex] is 2, so this is a quadratic function.
### Choice B: [tex]\( f(x)=2x^3+3x^2-5 \)[/tex]
This expression is already simplified:
[tex]\[
f(x) = 2x^3 + 3x^2 - 5
\][/tex]
The highest power of [tex]\( x \)[/tex] is 3, so this is a cubic function, not a quadratic function.
### Choice C: [tex]\( f(x)=5^2 \)[/tex]
Simplify this expression:
[tex]\[
f(x) = 25
\][/tex]
This is a constant function, not a polynomial in terms of [tex]\( x \)[/tex]. Therefore, it cannot be considered quadratic.
### Choice D: [tex]\( f(x)=\frac{2}{x} \)[/tex]
This expression represents a rational function:
[tex]\[
f(x) = \frac{2}{x}
\][/tex]
Here, [tex]\( x \)[/tex] is in the denominator, and the function is not a polynomial. It is not a quadratic function.
### Conclusion
Based on the above evaluations, the only quadratic function among the given choices is:
Choice A: [tex]\( f(x)=3(x-4)(x+3) \)[/tex]
