Two students are solving a quadratic equation of form x^2 + px + q = 0. One starts with a wrong value of p and finds the roots to be 2 and 6. The other starts with a wrong value of q and finds the roots to be 2 and -9. Find the correct roots of the equation.

We are told that the equation is of the form x^2 + px + q = 0.

The first guy, who has got the value of p wrong, solves the equation and finds out the roots to be 2 and 6. 

To understand the case in an easier and better way, lets assume that he thought the equation was x^2 + p'x + q = 0. The roots of this equation are 2 and 6. We can now deduce that in this case

Sum of roots = -p'/1 = 2 + 6 

=> p' = -8

Product of roots = q/1 = 12

=> q = 12

This gives us the value of q, which is 12. The first guy had got the value of the coefficient of p wrong, but he got the value of q right, which means that q = 12.

Similarly. the second guy, who has the wrong value of q, solves the equation, say, x^2 + px + q' = 0, and finds out the roots to be 2 and -9. So, we can say that

Sum of roots = -p = 2 + (-9) = -7 => p = 7

Product of roots = q' = 2 * -9 = -18

This guy has got the value of p right, so p = 7.

Hence, the original equation must be x^2 + 7x + 12 = 0, solving which we'd get the roots -4 and -3.