Question

Solve the Equation.

$x^2\oplus \left[8\right]⊝x=\left[0\right]$in${\mathrm{\mathbb{Z}}}_9$

Short answer

$x=\left[0\right]$and $x=\left[1\right]$are the solutions of $x^2\oplus 8⊝x=\left[0\right]$ in${\mathrm{\mathbb{Z}}}_9$ .

Step-by-step solution

Obtain solution

$x$	$x^2\oplus \left[8\right]⊝x$			Solution
$\left[0\right]$	$\left[0^2+8·0\right]$	$=\left[0\right]$	$=\left[0\right]$	Yes
$\left[1\right]$	$\left[1^2+8·1\right]$	$=\left[9\right]$	$=\left[0\right]$	Yes
$\left[2\right]$	$\left[2^2+8·2\right]$	$=\left[0\right]$	$=\left[2\right]$	No
$\left[3\right]$	$\left[3^2+8·3\right]$	$=\left[33\right]$	$=\left[6\right]$	No
$\left[4\right]$	$\left[4^2+8·4\right]$	$=\left[48\right]$	$=\left[3\right]$	No
$\left[5\right]$	$\left[5^2+8·5\right]$	$=\left[65\right]$	$=\left[2\right]$	NO

Result

The solution of $x^2\oplus 8⊝x=\left[0\right]$ in${\mathrm{\mathbb{Z}}}_9$ are$x=\left[0\right]$ and$x=\left[1\right]$ .