Discussion

If \(z = 2 + i, i^{2} = -1\) , then the value of \(z^{2} - 4z + 15\)

Answer: E

The value is 10.

Square \(z = 2 + i\) first, using \(i^{2} = -1\):

\[ \begin{aligned} z^{2} &= 4 + 4i + i^{2} = 3 + 4i \\ z^{2} - 4z + 15 &= (3 + 4i) - (8 + 4i) + 15 \\ &= 10 \end{aligned} \]

The imaginary parts cancel exactly. A quicker route: \(z - 2 = i\), so \((z-2)^{2} = -1\), that is \(z^{2} - 4z + 4 = -1\), and adding 11 gives 10.

15 is what you get by forgetting that \(z^{2} - 4z\) is not zero; 6 and 12 come from treating \(i^{2}\) as \(+1\) at some step.

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