AS 91577 · 2025 paper · a quadratic from one root, a surd squared, an equation in √x, a cubic with two unknowns, and simultaneous equations
Question 3(b) Achieved
Practises Arithmetic & the conjugate
Five separate parts: a quadratic built from one complex root, the square of a complex number written with surds, an equation in √x, a cubic with two unknown coefficients and one root given, and a pair of simultaneous equations in two complex unknowns.
Expand and simplify (√3a − √12a i)2, giving your answer in terms of a, where a is a real number.
Work it out
Try it on paper first. Stuck? Take one hint at a time.
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1 Square a bracket of two terms: the first squared, twice their product, the second squared.
(√3a)2 − 2√3a √12a i + (√12a i)2 -
2 Simplify each term. What is √3a × √12a, and what is i2?
For a ≥ 0, √3a √12a = √36a2 = 6a. And i2 = −1.
= 3a − 12ai − 12a -
3 Collect the real terms.
= −9a − 12ai. (If a is negative, both roots are imaginary and the working changes, but the answer comes out the same.)
What the marker wanted
- Achieved The expression fully simplified.
In our words, from NZQA’s 2025 marking schedule.
How did it go?
Got it moves it further out. Review brings it back tomorrow.