Solve quadratics by inspection, square roots, completing the square, formula, and factoring; express complex solutions.
Solve a quadratic equation with complex solutions by completing the square and writing the answers using i
Problem
Solve \(x^{2}+9=0\) over the complex numbers. Isolate \(x^{2}\), rewrite the square root of \(-1\) using \(i\), and include both square-root branches.
Big Picture
What this problem is really about
After isolating the square, the negative value on the other side tells us the solutions are complex rather than real. We’ll take both square-root branches, separate the negative radicand into a positive square times negative one, replace the square root of negative one with i, and verify both results by squaring.
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