Ask Uncle Colin: Three Unknowns

Dear Uncle Colin,

I’ve been told that $(ax+b)(bx+a)\equiv 10x^2+cx+10$, with $a$ and $b$ positive integers, and I need to find the possible values for $c$. How?!

- This Ridiculous Identity Puzzle Looks Easy

Hi, TRIPLE, and thanks for your message!

The first thing I would do would be to expand the left hand side and try to match coefficients.

• $ab{x}^2+(a^2+b^2)x+ab\equiv$10x^2 + cx + 10$

That means that $ab=10$ and $a^2+b^2=c$.

That’s not enough to solve for $c$ in the reals – we have two equations in three unknowns – but there aren’t many possibilities for $a$ and $b$ in the positive integers.

There are only two possible pairs of answers for $a$ and $b$: they’re either $1$ and $10$ in either order, or they’re $2$ and $5$ in either order.

If they’re $1$ and $10$, then $c=1^2+{100}^2=101$.

If they’re $2$ and $5$, then $c=2^2+5^2=29$.

So the possible values for $c$ are 101 and 29.

Hope that helps!

- Uncle Colin