Two students A and B solve an equation of the from x 2 + px + q = 0. A starts with a wrong value of p and obtains the roots as 2 and 6. B starts with a wrong value of q and gets the roots as 2 and -9. What are the correct roots of the equation?

Let α and β be the roots of the quadratic equation x² + px + q = 0
Given that, A starts with a wrong value of p and obtains the roots as 2 and 6. But this time q is correct. i.e products of roots
= q = α . β = 6 x 2 = 12 ...(i)
and B starts with a wrong value of q and gets the roots as 2 and - 9. But this time p is correct i.e., sum of roots
= p = α + β = - 9 + 2 = - 7 ..(ii)
(α - β)² = (α + β)² - 4α β
= (-7)² - 4.12 = 49 - 48 = 1
[from Eqs. (i) and (ii)]
⇒ α - β = 1 ..(iii)
Now, from Eqs. (ii) and (iii) , we get
α = - 3 and β = - 4
which are correct roots .