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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 $abc$.

 Sep 1, 2023
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\(x(x-3)=1+8x-5\)

Expand and simplify:

\(x^2-3x=1+8x-5 \\ \iff x^2-11x+4=0\)

\(\text{Now, use the quadratic formula: } \) \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

\(\text{Where: a=1,b=-11,c=4}\)

Try to continue and if you are stuck, please do not hesitate to ask for further help :)

 Sep 2, 2023

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