193245 Two candidates attempt to solve a quadratic of the form $\displaystyle x^{2}+px+q= 0.$ 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.

The first candidate uses the correct value of q, so the product of their incorrect roots gives the actual q, which is 2 times 6 equals 12. The second candidate uses the correct value of p, so the sum of their incorrect roots gives the actual p, which is 2 plus negative 9 equals negative 7. The correct equation is therefore x squared plus 7 x plus 12 equals 0. Factoring this quadratic equation gives x plus 3 times x plus 4 equals 0, meaning the correct roots are negative 3 and negative 4.

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