does $(f(x)+k)^2$ = $f(x)^2+2kf(x)+k^2$

Question

if you have a function $(f(x)+k)^2$ where $k>0$ is this the same as $f(x)^2+2kf(x)+k^2$? It should be due to quadratic expansion or not?

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Answer 1

Yes. This is true. Take:$$f(x)=10x-1$$$$k=2$$

$$(10x-1+2)^2=10x+1=100x^2+20x+1$$

$$(10x-1)^2+2(2)(10x-1)+(2)^2=100x^2-20x+1+40x-4+4=100x^2+20x+1$$

Both ways, the outcome is $100x^2+20x+1$.

Answer 2

The answer is yes and $ k$ does not have to be positive.

For example if $f(x)=x^3$ and $k=-2$ , then $$(f(x)+k)^2 =(x^3-2)^2 = x^6 - 4x^3+4 = f(x) ^2 +2kf(x)+ k^2$$

Notice that for a real number x, f(x) is a real number and it stisfies all properties of real numbers.