Solve real-coefficient quadratic equations that have complex solutions.
Solve a real-coefficient quadratic with complex solutions by completing the square
Problem
Solve \((x-3)^{2}=-16\) by taking \(\text{both}~\text{complex}~\text{square}~\text{roots}\), obtaining \(x-3=\pm~4i\), and isolating \(x\).
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What this problem is really about
The equation already presents an isolated squared binomial, so no expansion is needed. We’ll take both complex square roots, rewrite the negative radical with i, undo the horizontal shift with the correct sign, and verify each conjugate solution in the original square.
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