AS 91577 · 2023 paper · a squared surd, no real roots, a real part of zero, a complex cubic, and a modulus equation
Question 1(b) Achieved
Practises Polynomial equations with complex roots
Five separate parts: a bracket with a surd squared, a quadratic with no real roots, a proof from a real part of zero, a cubic with one complex root given, and a constant found from a modulus.
Find the value(s) of r so that the quadratic equation 4x2 − 4x + 3r − 2 = 0 has no real roots.
Work it out
Try it on paper first. Stuck? Take one hint at a time.
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1 No real roots means what about the discriminant?
b2 − 4ac < 0, with a = 4, b = −4 and c = 3r − 2. -
2 Write it out.
(−4)2 − 4(4)(3r − 2) < 0
16 − 48r + 32 < 0
48 − 48r < 0 -
3 Solve the inequality. Careful when you divide by a negative.
−48r < −48. Dividing by −48 turns the inequality round: r > 1.
What the marker wanted
- Achieved The correct inequality for r.
In our words, from NZQA’s 2023 marking schedule.
How did it go?
Got it moves it further out. Review brings it back tomorrow.