Level 3 Complex Numbers Walkthrough
2025 NCEA Level 3 Complex Numbers Question 1(c)
2025 Paper
Question
Prove that the equation has real roots for all real values of \(k\), where \(k \neq 0\).
First walkthrough idea
Focus to try first
turning the equation into a quadratic in \(x\), finding its discriminant, and justifying why it is always positive.
Step 1
Spot the proof idea
If the discriminant is positive, the quadratic has real roots.
Show the first step’s working
If the discriminant is positive, the quadratic has real roots.
Key result
A positive discriminant.Walkthrough overview
What this question practises
This 2025 walkthrough is part of AS91577 — Apply the algebra of complex numbers in solving problems.
Method: Discriminants and proving real roots for every real \(k\).
This is Question 1(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.
Page updated .
Learning summary
Review the method, not only the answer
This walkthrough helps you practise discriminants and proving real roots for every real \(k\). 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.