a quadratic equation is of the form
[tex]y=ax^2+bx+c[/tex]
then. if x = 0, y = 7:
[tex]\begin{gathered} 7=a(0)^2+b(0)+c \\ c=7 \end{gathered}[/tex]
and, if x = 1, y = 16
[tex]\begin{gathered} 0=a(1)^2+b(1)+7 \\ 0=a+b+7 \\ a+b=-7\text{ eq 1 } \end{gathered}[/tex]
and if x = 2, y = 27
[tex]\begin{gathered} 27=a(2)^2+b(2)+7 \\ 27=4a+2b+7 \\ 27-7=4a+2b+7-7 \\ 4a+2b=20\text{ } \\ 2a+b=10\text{ eq2} \end{gathered}[/tex]
then solve for a and b with the equations 1 and 2
[tex]\begin{gathered} \begin{bmatrix}a+b=-7 \\ 2a+b=10\end{bmatrix} \\ a+b=-7 \\ a+b-b=-7-b \\ a=-7-b \\ 2\mleft(-7-b\mright)+b=10 \\ -14-2b+b=10 \\ -14-b=10 \\ -14-b+14=10+14 \\ -b=24 \\ \frac{-b}{-1}=\frac{24}{-1} \\ b=-24 \end{gathered}[/tex]
for a
[tex]a=-7-b=-7-(-24)=-7+24=17[/tex]
answer, the equation is:
[tex]y=17x^2-24x+7[/tex]
