# Can there be two complex roots for the square root of a negative number? [closed]

Can the Square root of (-9) have two answers namely +3i or -3i?

Squaring both 3i and -3i does give -9.

Why does my text book say only 3i?

## closed as off-topic by Hans Lundmark, Simply Beautiful Art, Leucippus, Shailesh, Xander HendersonOct 9 '17 at 3:48

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• Both $3i$ and $-3i$ are valid solutions, you're right. The reason they might ignore the negative solution is possibly that they're only considering what's called the "principal" solution ($3i$) and they either think it's implied or irrelevant that $-3i$ is also a solution. But nevertheless, you're right that there are two solutions. – Jam Oct 8 '17 at 14:39
• How does your textbook define the square root? The equation $x^2 = -9$ has the two solutions you gave. – gammatester Oct 8 '17 at 14:40
• @gammatester -9 nested in a Square Root(symbol) – pirsquare Oct 8 '17 at 14:50
• That's no definition. – gammatester Oct 8 '17 at 14:51
• @gammatester sorry I misread your question. What I meant is the question was not in x^2=-9 form – pirsquare Oct 8 '17 at 14:53

I don't know why your text says that. Perhaps that the author is just saying that $3i$ is a square root of $-9$, without saying that it is the only square root.
More generally, every non-zero complex number $w$ has two distinct square roots. If $r$ is one such root, then $-r$ is another square root and, since $r\neq0$, $r\neq-r$. Since the equation $z^2-w=0$ can have no more than two roots, there can be no other root.