Porisma

Worked solution

Reuse a known square after multiplying by i

Express (3−i)2(3-i)^2 in the form a+iba+ib. Hence or otherwise find the roots of (z+i)2=−8+6i(z+i)^2=-8+6i. marks

[N04/I/12(a)]

CJC · Review problems · Question 6

Answer

(3−i)2=8−6i(3-i)^2=8-6i; z=1+2iz=1+2i or z=−1−4iz=-1-4i.

Find the given square

(3−i)2=9−6i+i2=8−6i(3-i)^2=9-6i+i^2=8-6i.

How did this part go?

Change its sign and take both roots

−8+6i=−(3−i)2=[i(3−i)]2=(1+3i)2-8+6i=-(3-i)^2=[i(3-i)]^2=(1+3i)^2. Thus z+i=±(1+3i)z+i=\pm(1+3i) and z=±(1+3i)−iz=\pm(1+3i)-i. The two roots are z=1+2iz=1+2i and z=−1−4iz=-1-4i. Squaring either shifted value (z+i)(z+i) returns −8+6i-8+6i.

How did this part go?

Common mistakes

Apply ± to the whole root 1+3i before subtracting i. The translation −i is applied after choosing each sign.

Solution notes

Source filename says 2026, but its printed header says 2025/JC2; both are retained without silently changing the source identity.

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