## Two Prime Roots

Two prime numbers are roots of the quadratic equation $x^2-63x+k=0$. What value of $k$ makes this a true statement?
Source: mathcontest.olemiss.edu 11/13/2006

SOLUTION
Let $a$ and $b$ be two prime numbers that are roots of the quadratic equation $x^2-63x+k=0$.
We factor the equation as follows:
$x^2-63x+k=\left (x-a\right )\left (x-b\right )$
$x^2-63x+k=x^2-bx-ax+ab$
develop the right hand side
$x^2-63x+k=x^2-\left (a+b\right )x+ab$  simplify

Compare the left and right hand side of the above equation
$a+b=63$ and $ab=k$

Try a few prime numbers
$a=2;\;b=61$  YES a solution
$a=3;\;b=60$  NOT a solution because is not prime
$a=5;\;b=58$  NO
$a=7;\;b=56$  NO
$\cdots$
Thus, $k=ab=2\times 61=122$

Answer: $122$.