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Find all the solutions of $$x^2 + 5x - 9 \equiv 0\mod5^2$$

I know that first I must find the solutions of $f(x)\equiv0\mod 5$ and then "get up" those solutions modulo $5^2$. So $$f(x)\equiv x^2+1 \equiv0\mod 5\quad \Rightarrow\quad x\equiv 2, x\equiv -2\mod 5$$

But from here on I don't know how to handle the solutions modulo $5$ to get the solutions modulo $5^2$. Any help with proper justifications would be apreciated.

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    $\begingroup$ Ok, so you know that $x\equiv \pm 2 \pmod 5$, so write $x=\pm 2 +5t$ and solve for $t$. Alternatively, just remark that there aren't very many residues $\pmod {25}$ which are $\pm 2 \pmod 5$ so you could just try each one. $\endgroup$
    – lulu
    Commented May 14, 2017 at 12:13

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$$x^2+5x-9\equiv2(x^2+5x-9)\equiv0\mod25$$ Now by calculation $x=7$ satisfies the condition. So $x-7 $divides $ 2(x^2+5x-9)$ We have $$2(x^2+5x-9)\equiv(x-7)(2x+24)\equiv0\mod25$$ $$x\equiv7\mod25$$ and $$x\equiv13\mod25$$

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