Determining the coefficients of the quadratic functions
1 Answer
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
\[f(x)=2x^2+\frac{23}{15}x-6 \text{and} g(x)=\frac{4}{15} x^2+2x-4.\]
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