Solve real-coefficient quadratic equations that have complex solutions.
Find a quadratic equation with real coefficients from a pair of complex conjugate roots
Problem
Find a quadratic equation with real coefficients whose roots are \(2+3i\) and \(2-3i\).
Big Picture
What this problem is really about
Each root supplies a linear factor, and conjugate roots guarantee the imaginary terms disappear. We’ll form both factors, expose their conjugate structure around the common real part, apply the conjugate product, reduce the imaginary square, and expand to a real-coefficient quadratic.
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