Question

The roots of the quadratic equation $x(x-3)=1+8x-5$ may be expressed in the form $\frac{a+\sqrt{b}}{c}$ and $\frac{a-\sqrt{b}}{c}$, where $a$, $b$, and $c$ are prime numbers. Find $abc$.

Answer 1

First we simplify to $x^2-11x+6 = 0$. Now we solve for $x$ using $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ (note that $a=1$, $b=-11$, and $c=6$). The roots of this equation are
\[x = \frac{11\pm \sqrt{11^2-24}}{2}=\frac{11\pm \sqrt{97}}{2}.\]
Therefore $a=11$, $b=97$, and $c=2$, so our answer is $abc=\boxed{2134}$.