Does Prime Matter?

Algebra Level 3

The quadratic equation $x^2 + ax + b + 1 = 0$ , where $a$ and $b$ are integers, has positive, integer-valued roots.

Is it possible that $a^2 + b^2$ is a prime number?

Yes No

1 solution

Jason Chrysoprase
Dec 26, 2016

Assume that $p$ and $q$ are the roots of the equation

By using Vieta's Formula, we know that $p+q = -a$ and $pq = b+1$

Rearrange the expression

$\begin{aligned} a^2 + b^2 & = {\left(- {\left(p+q \right)} \right)}^2 + {\left(pq - 1\right)}^2 \\ &= p^2 q^2 + p^2 + q^2 + 1 \\ &= {\left(p^2 + 1\right)} {\left(q^2 + 1\right)} \\ \end{aligned}$

Remember that a prime number has factor of $1$ and the number itself, but ${\left(p^2 + 1\right)}$ or ${\left(q^2 + 1\right)}$ can't have the value of $1$ since $p$ and $q$ are non-zero integers number as in the problem's description.

Therefore, $a^2 + b^2$ will not get prime number

Can you add a line to explain why $p, q$ are non-zero numbers? Thanks!

Calvin Lin Staff - 4 years, 5 months ago

Sure, it's was mentioned on the problem description that the roots are positive non-zero integers

Jason Chrysoprase - 4 years, 5 months ago

Right. The idea is to present all of the information in the solution so that the reader can easily see why your conclusions immediately follow, as opposed to expecting them to have recalled various facts that were stated somewhere in the problem statement, especially if they were not able to solve the problem.

Calvin Lin Staff - 4 years, 5 months ago

Does Prime Matter?

1 solution

0 pending reports