15. Let $x_{2}-x_{1}=x_{3}-x_{2}=\cdots=x_{7}-x_{6}=d$, and the function values corresponding to $x_{i}$ be $y_{i}$. Then $\Delta_{k}=y_{k+1}-y_{k}=$ $\left(a x_{k+1}^{2}+b x_{k+1}+c\right)-\left(a x_{k}^{2}+b x_{k}+c\right)=a\left[\left(x_{k}+d\right)^{2}-x_{k}^{2}\right]+b\left[\left(x_{k}+d\right)-x_{k}\right]=2 a d x_{k}+\left(a d^{2}+b d\right)$. Therefore, $\Delta_{k+1}-\Delta_{k}=2 a d\left(x_{k+1}-x_{k}\right)=2 a d^{2}$ (a constant). From the given data
\begin{tabular}{lcccccccc}
$y_{k}:$ & 51 & 107 & 185 & 285 & 407 & 549 & 717 \\
$\Delta_{k}:$ & & 56 & 78 & 100 & 122 & 142 & 168 \\
$\Delta_{k+1}-\Delta_{k}:$ & & 22 & 22 & 22 & 20 & 26
\end{tabular}

It can be seen that 549 is the incorrect $y$ value, and its correct value should be 551.
15. A student, in order to plot the graph of the function $y=a x^{2}+b x+c(a \neq 0)$, took 7 values of the independent variable: $x_{1}<$ $x_{2}<\cdots<x_{7}$, and $x_{2}-x_{1}=x_{3}-x_{2}=\cdots=x_{7}-x_{6}$, and calculated the corresponding $y$ values. The following table is listed.
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline$x$ & $x_{1}$ & $x_{2}$ & $x_{3}$ & $x_{4}$ & $x_{5}$ & $x_{6}$ & $x_{7}$ \\
\hline$y$ & 51 & 107 & 185 & 285 & 407 & 549 & 717 \\
\hline
\end{tabular}

But due to carelessness, one of the $y$ values was calculated incorrectly. Please point out which value is incorrect and what the correct value should be. Explain your reasoning.