Let the correct equation be:

$ax^2 + bx + c = 0$

For the first student, *b* is wrong but *a* and *c* are correct. Thus, the product of roots is the same as that of the correct equation.

$x_1 x_2 = \dfrac{c}{a}$

$3(-2) = \dfrac{c}{a}$

$\dfrac{c}{a} = -6$

For the second student, *c* is wrong but *a* and *b* are correct. Thus, the sum of roots is the same as that of the correct equation.

$x_1 + x_2 = -\dfrac{b}{a}$

$3 + 2 = -\dfrac{b}{a}$

$\dfrac{b}{a} = -5$

From the correct equation

$ax^2 + bx + c = 0$

$x^2 + \dfrac{b}{a}x + \dfrac{c}{a} = 0$

$x^2 - 5x - 6 = 0$ ← the correct equation

Solving for the correct roots

$x^2 - 5x - 6 = 0$

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

$x = 6 \, \text{ and } \, -1$ ← the correct roots (*answer*)