Understand i as a number with i^2=-1 and represent complex numbers as a+bi.
Solve equations of the form \(x^2 =\) a negative number using \(i\)
Problem
Solve \(x^{2}=-1\) over the complex numbers by taking \(\text{both}~\text{square}~\text{roots}\). State the complete solution set and verify that each value squares to \(-1\).
Big Picture
What this problem is really about
Solving a square equation requires both square-root branches, even when the radicand is negative. We’ll take plus and minus the square root, rewrite the negative root using the imaginary unit, verify each candidate by squaring, and report the complete complex solution set.
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