I need to show that if $p$ is an odd prime not dividing $ac$, then complete solutions to $$ax^2 +bx +c \equiv 0 \pmod p $$and $$cx^2 + bx + a \equiv 0 \pmod p$$

have the same number of solutions.

I also need to show that if $ab \equiv 1 \pmod p$, $p$ an odd prime, then $$\left(\frac{a}{p}\right) = \left(\frac{b}{p}\right)$$

(The Legendre symbols are equal)


Hints: (1) The roots are invertible modulo $p$. Let $r$ be a root of the first congruence, and let $rs\equiv 1\pmod{p}$. So $s$ is the modular inverse of $r$. By multiplying the first congruence through by $s^2$, show that $s$ is a root of the second congruence.

(2) Suppose that $w^2\equiv a\pmod{p}$. Then $b^2w^2\equiv b\pmod{p}$.

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