One student make a mistake in copying the coefficient of x and got a root of +3 and -2. Another student made a mistake in copying the constant term and got a root of +3 and +2. Find the correct equation and correct roots.

Solution:
Let the correct equation be:

Ax^2 + Bx + c

and the correct roots be:

x1 and x2

Consider the first student's equation. It is wrong in B but correct in A and C. Thus,product of roots is the same as that of the correct equation.

C/A = 3(-2)= -6

Consider the 2nd student's equation. It is wrong in C but correct in A&B.Thus,sum of roots is the same as that of the correct equation.

-B/A = 3 + 2

B/A = -5

Consider the correct equation:

Ax^2 + Bx + c = 0 ------> divide by A

x^2 + B/Ax + C/A =0

thus

x^2-5x-6=0 -------> correct equation

solve for roots by factoring:

x^2-5x-6=0

(x-6)(x+1)=0

x=6 and -1

Therefore,the correct roots are:

6 and -1