## Discussion Forum

### Daily Magic Spell Discussion - January 23, 2017

Daily Magic Spell Discussion - January 23, 2017

For today, we will discuss about the Daily Magic Spell problem given on January 23, 2017.

Problem: Given that $2+ai$ and $b+i$ are the two roots of the quadratic equation $x^2+px+q=0$ where $p$ and $q$ are real numbers.  What is $q$?

Solution: By Viete's formulas, $p=-(2+ai + b + i)=-(b+2)-(a+1)i$, and $q=(2+ai)(b+i) = (2b-a)+(2+ab)i$.  Since $p$ and $q$ are real numbers, $a+1=0$ and $2+ab=0$.  Thus $a=-1$, $b=2$ and therefore $p=-4, q=5$.

Some common methods that could be used to tackle this problem by a student who have incorrectly answered the question were guess and check and using brute force to determine the values of $p$ and $q$.

The problem by doing this is that it doesn't allow you to determine the solution in a very efficient way. One should take advantage of using Viete's formulas to get the solution to this problem.