How to solve Quadratic Equations with an Unknown C and other variables. For instance $3x^2 - 11x + r$, I understand the value of $r$ is $6$ through trial and error but trial and error is extremely inefficient and time consuming thus not useful in exam situations, How would I solve this?
And also : If $x^2 +px + q$ is a factor of $x^3 + ax^2 + b$,
prove that:
$b = q(a-p)$

A:  $3x^2-11x +r=0$
The  codition is that discreminet $\Delta\geqslant 0$, so we must have:
$\Delta= 11^2-12 r\geqslant0$
Or: $r\leqslant 10$
Now $121$ is odd and $12r$ is even and we have to find odd perfect squares less than $121$ which are $81, 49, 25$ and $9$. Only $49$ and $25$ give integers for $r$:
$r= 121-49=72=12r$ so $r=6$
$r=121-25=96=12r$ so $r=8$
B: $x^2+px+q|x^3+ax+b$
By direct division you find that thr remainder is $(a-p)x^2-qx+b$ which must be equal to zero, so we have:
$x[(a-p)x-q]=-b$
Therefore b must divide both $q$ and $(a-p)$ because $(a-p)$ and $q$ have no common divisors, hence we must have $b|q(a-p)$
