Level 3 Complex Numbers Walkthrough

2025 NCEA Level 3 Complex Numbers Question 3(c)

2025 Paper

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Question

Solve the following equation for \(x\).

\[ (1+2\sqrt{x})(3+2\sqrt{x})=5+6\sqrt{x} \]

First walkthrough idea

Focus to try first

expanding a radical equation, using a substitution, and checking the valid solution.

Step 1

Make the substitution

That turns the radicals into an ordinary quadratic in \(u\).

Show the first step’s working

That turns the radicals into an ordinary quadratic in \(u\).

Key result

\(\,u=\sqrt{x}\)

Walkthrough overview

What this question practises

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

Method: Solving a radical equation with the substitution \(\sqrt{x}=u\).

This is Question 3(c) 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 solving a radical equation with the substitution \(\sqrt{x}=u\). Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

State or check the real-domain restriction and test for extraneous solutions after squaring.

Continue practising

← Question 3(b) Question 3(d) →