Question

If 4f(x)+f(5-x)=x2 What is f(x)?

Answer 1

We are given that $4f(x)+f(5-x)=x^2$.

Substitute x with 5-x, then the above equation becomes:

$4f(5-x)+f(5-(5-x))=(5-x)^2$, that is

$4f(5-x)+f(x)=(5-x)^2$


So, we have the following system of equations:

i) $4f(x)+f(5-x)=x^2$
ii) $4f(5-x)+f(x)=(5-x)^2$

multiply the first equation by -4, so that we eliminate f(5-x)'s

i) $-16f(x)-4f(5-x)=-4x^2$
ii) $4f(5-x)+f(x)=(5-x)^2$

adding the 2 equations side by side we have:

$-15f(x)=-4x^2+(5-x)^2$

expanding the binomial, and collecting same terms we have:

$-15f(x)=-4x^2+(25-10x+x^2)$

$-15f(x)=-3x^2-10x+25$

dividing by -5:

$3f(x)=\frac{3}{5}x^2+2x-5$

dividing by 3:

$\displaystyle{f(x)= \frac{1}{5}x^2+ \frac{2}{3}x-\frac{5}{3}$


Answer: $\displaystyle{f(x)= \frac{1}{5}x^2+ \frac{2}{3}x-\frac{5}{3}$