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(a) Achieved
Practises Polynomial equations with complex roots
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.
One solution of a quadratic equation, with real coefficients, is x = 2 + √p i. Find the quadratic equation, in terms of p, giving your answer in the form ax2 + bx + c = 0.
Work it out
Try it on paper first. Stuck? Take one hint at a time.
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1 The coefficients are real. What is the other root?
Its conjugate, x = 2 − √p i. -
2 Multiply the two factors together.
(x − 2 − √p i)(x − 2 + √p i) = (x − 2)2 − (√p i)2 = (x − 2)2 + p -
3 Expand into the form asked for.
x2 − 4x + 4 + p = 0 -
4 A quicker route, as a check?
x − 2 = √p i. Squaring gives x2 − 4x + 4 = −p, the same equation.
What the marker wanted
- Achieved The quadratic equation.
In our words, from NZQA’s 2025 marking schedule.
How did it go?
Got it moves it further out. Review brings it back tomorrow.