Solve real-coefficient quadratic equations that have complex solutions.
Verify that a given complex number is a solution of a quadratic equation
Problem
Test \(x=2+3i\) in \(x^{2}-4x+13\). Expand \((2+3i)^{2}\) including the middle term, substitute \(i^{2}=-1\), combine \(\text{all}~\text{terms}\), and state whether the \(\text{result}~\text{is}~\text{zero}\).
Big Picture
What this problem is really about
Verification means substituting the candidate everywhere and showing that both complex components cancel. We’ll expand the binomial square including its middle term, reduce i squared, distribute the linear coefficient, combine real and imaginary parts separately, and test whether the polynomial becomes zero.
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