Solve real-coefficient quadratic equations that have complex solutions.
Solve a quadratic of the form ax^2 + c = 0 over the complex numbers by using square roots
Problem
Solve \(x^{2}+25=0\) over the complex numbers by isolating \(x^{2}=-25\), taking \(\text{both}~\text{square}~\text{roots}\), and stating the complete solution set.
Big Picture
What this problem is really about
The missing linear term makes the square-root method direct. We’ll isolate the squared variable, take both square-root branches, separate the negative radicand into a positive square and the imaginary unit, and substitute both conjugate values back to verify the complete set.
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