What is the procedure to solving $x^2$ $mod$ $23 = 7^2$? According to WolframAlpha, there is no integer solution but I am completely confused as to what steps was taken to determine that.
Before asking the question, I did try to solve this using brute force by plugging in some arbitrary numbers to see whether the square of that number mod $23$ gave me a remainder of $49$ but the procedure was quite tedious. Hence the reason I turned to WolframAlpha.
Going back to the question, I want to know how WoflramAlpha determined that there was no integer solution to solving for $x$.
I'm not sure what WolframAlpha did, and it would be helpful if you posted a link to the result of your input. However, it should be fairly obvious that there is at least one solution to this equation, namely, if $x \equiv 7 \pmod{23}$, certainly $x^2 \equiv 7^2 \pmod{23}$.
It turns out you have the additional solution of $x \equiv -7 \equiv 16 \pmod{23}$, and that's it. In fact, whenever you're solving a quadratic equation modulo a prime, you will have at most two roots, sometimes one, and sometimes none.