smc

Let $x^2+bx+c=0$ represent the correct equation. Since the coefficient of the $x^2$ term is $1$, the sum of the roots is $-b$, and the product of the roots is $c$. If a student only misreads the constant term, he must have the correct sum of roots. Therefore, the sum of the roots is $8+2=10$, so $b=-10$. If a student only misreads the linear term, he must have the correct product of the roots. The product of the roots is $(-9)\cdot (-1)=9$, so $c=9$. The correct equation is $\boxed{x^2-10x+9=0}$.