Level 3 Complex Numbers Walkthrough

2025 NCEA Level 3 Complex Numbers Question 3(b)

2025 Paper

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Question

\[ \left(\sqrt{3a}-\sqrt{12a}\,i\right)^2 \]

Expand and simplify, giving your answer in terms of \(a\), where \(a\) is a real number.

First walkthrough idea

Focus to try first

squaring a binomial with a complex term and keeping track of the effect of \(i^2=-1\).

Step 1

Identify the expansion rule

Here the second term already contains the negative sign, so this rule still fits.

Show the first step’s working

Here the second term already contains the negative sign, so this rule still fits.

Key result

\(\,(A+B)^2=A^2+2AB+B^2\)

Walkthrough overview

What this question practises

This 2025 walkthrough is part of AS91577 — Apply the algebra of complex numbers in solving problems.

Method: Squaring and simplifying a complex expression.

This is Question 3(b) from the 2025 NCEA Level 3 Complex Numbers paper for AS91577 — Apply the algebra of complex numbers in solving problems. Use the guided hints to practise the method before revealing the full working.

Calc.nz is an independent learning resource. Compare questions, diagrams, and assessment information with the official NZQA resources for AS91577.

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Learning summary

Review the method, not only the answer

This walkthrough helps you practise squaring and simplifying a complex expression. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Check each step against the original condition, preserve signs and restrictions, and confirm that the final result answers the question asked.

Continue practising

← Question 3(a) Question 3(c) →