Level 3 Complex Numbers Walkthrough

2023 NCEA Level 3 Complex Numbers Question 1(a)

2023 Paper

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Question

\[ \text{Write }(5-2\sqrt{p})^2\text{ in the form }a+bp+c\sqrt{p}\text{ where }a,\ b,\ \text{and }c\text{ are integers.} \]

First walkthrough idea

Focus to try first

expanding a binomial square and collecting the constant, \(p\), and \(\sqrt{p}\) terms into the requested form.

Step 1

Identify the expansion rule

We square the first term, subtract twice the product, then add the square of the second term.

Show the first step’s working

We square the first term, subtract twice the product, then add the square of the second term.

Key result

\((a-b)^2=a^2-2ab+b^2\)

Walkthrough overview

What this question practises

This 2023 walkthrough is part of AS91577 — Apply the algebra of complex numbers in solving problems.

Method: Expanding a surd binomial into the form \(a+bp+c\sqrt{p}\).

This is Question 1(a) from the 2023 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 expanding a surd binomial into the form \(a+bp+c\sqrt{p}\). Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Keep exact values until the end and check any denominator or domain restriction.

Continue practising

Question 1(b) →