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

Complex roots of quadratic equations Problem set B Answers

Skills

    1. (a) Vertex \((-1, 2)\)
    2. (b) \(x = -1 \pm \frac{\sqrt{6}}{3}i\) (3 s.f.: \(-1 \pm 0.816i\))
    3. (c) \((-1, 0.816)\) and \((-1, -0.816)\). The real part equals the x-coordinate of the vertex.
    1. (a) \(x = \pm 8i\)
    2. (b) \(x = -4 \pm 2i\)
    3. (c) \(x = - \frac{3}{4} \pm \frac{\sqrt{31}}{4}i\), which is \(-0.75 \pm 1.39i\) to 3 s.f.
    1. (a) \(4 + 3i\)
    2. (b) \((4, -3)\) and \((4, 3)\)
    3. (c) 6 units

In context

    1. (a) Vertex \((4, 3)\): the drone is closest to the buoy, 3 m away, after 4 seconds
    2. (b) \(t = 4 \pm 2.45i\)
    3. (c) There is no real time at which the distance is 0, so the drone never reaches the buoy

Stretch

    1. (a) \(-3 - 4i\)
    2. (b) \(b = 6,\ c = 25\)
    1. (a) \(x = -4\)
    2. (b) \(k > 16\)
    3. (c) \(x = -4 \pm 5i\); distance 10

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