# Determining the coefficients of the quadratic functions

Let 𝑓(𝑥)= $2x^{2}$ + 𝑚𝑥 − 6 and 𝑔(𝑥)= $nx^{2}$ + 2𝑥 − 4. The functions are combined to form the new function h(𝑥) = 𝑓(𝑥) − 𝑔(𝑥). Points (− 3, 17) and (2, 4) satisfy the new function. Determine 𝑓(𝑥) and 𝑔(𝑥). Leave the final answer in exact form.

Let
$h(x)=f(x)-g(x)=(2x^2 + 𝑚𝑥 − 6) -(nx^2+2x-4)=(2-n)x^2+(m-2)x -2.$
We have
$17=h(-3)=9(2-n)-3(m-2)-2=-9n-3m+24.$
$\Rightarrow 9n+3m=7. (1)$

Also
$4=h(2)=4(2-n)+2(m-2)-2=-4n+2m+2$
$\Rightarrow 2n-m=-1. (2)$
Multiply equation (2) by 3 and add it to equation (1) to get
$15n=4 \Rightarrow n=\frac{4}{15}.$
Substituting in (2) we get
$m=2n+1=\frac{8}{15}+1=\frac{23}{15}.$
Hence
$f(x)=2x^2+\frac{23}{15}x-6 \text{and} g(x)=\frac{4}{15} x^2+2x-4.$

Erdos
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