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IB Mathematics: Applications and Interpretation HL · AHL 6: Complex Numbers · AHL 6.4

Complex roots of quadratic equations Problem set A Answers

Skills

    1. (a) Vertex \((2, 3)\)
    2. (b) \(x = 2 \pm \sqrt{3}i\) (3 s.f.: \(2 \pm 1.73i\))
    3. (c) \((2, 1.73)\) and \((2, -1.73)\). The real part equals the x-coordinate of the vertex.
    1. (a) \(x = \pm 7i\)
    2. (b) \(x = 3 \pm 4i\)
    3. (c) \(x = - \frac{1}{3} \pm \frac{\sqrt{11}}{3}i\), which is \(-0.333 \pm 1.11i\) to 3 s.f.
    1. (a) \(2 - 5i\)
    2. (b) \((2, 5)\) and \((2, -5)\)
    3. (c) 10 units

In context

    1. (a) Vertex \((3, -3)\): the best the stall can do is a loss of 3 thousand euros, at a price of 3 euros
    2. (b) \(x = 3 \pm 1.22i\)
    3. (c) There is no real price at which the profit is 0, so the stall never breaks even

Stretch

    1. (a) \(2 - 5i\)
    2. (b) \(b = -4,\ c = 29\)
    1. (a) \(x = 3\)
    2. (b) \(k > 9\)
    3. (c) \(x = 3 \pm 3i\); distance 6

jmaths.xyz · AI HL · AHL 6.4 Complex roots of quadratic equations