Level 3 Complex Numbers Walkthrough

2025 NCEA Level 3 Complex Numbers Question 3(a)

2025 Paper

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Question

One solution of a quadratic equation, with real coefficients, is

\[ x = 2 + \sqrt{p}\,i \]

Find the quadratic equation, in terms of \(p\), giving your answer in the form \(ax^2+bx+c=0\).

First walkthrough idea

Focus to try first

using the conjugate-root rule for real coefficients and building the quadratic equation from the two roots.

Step 1

Find the other root

Real-coefficient quadratics have complex roots in conjugate pairs.

Show the first step’s working

Real-coefficient quadratics have complex roots in conjugate pairs.

Key result

\(\,x=2-\sqrt{p}\,i\)

Walkthrough overview

What this question practises

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

Method: Conjugate roots and forming a quadratic with real coefficients.

This is Question 3(a) 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 conjugate roots and forming a quadratic with real coefficients. 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 2(e) Question 3(b) →