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AI HL · AHL 6

Complex Numbers

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AHL 6.4

Complex roots of quadratic equations

AHL 6.4

Complex roots of quadratic equations

Target: I can use technology to graph a quadratic and read the coordinates of its vertex. (+4 more)

By the end of this section

Key points

No real roots, two complex roots

  • A quadratic \(ax^2 + bx + c = 0\) with \(b^2 - 4ac < 0\) has no real roots. Its graph does not cross the x-axis.
  • It still has two roots, and they are complex. For \(x^2 + 9 = 0\), \(x = \pm 3i\).
  • Put the calculator in complex mode before you use the solver. In real mode it reports no solution.
  • Give the roots in the form \(x + iy\), exact where you can and otherwise to 3 significant figures.

Conjugate pairs and the vertex

  • If a quadratic has real coefficients, its complex roots are a conjugate pair: \(p + qi\) and \(p - qi\).
  • On an Argand diagram the two roots are reflections of each other in the real axis.
  • The real part \(p\) is the x-coordinate of the vertex, \(-\dfrac{b}{2a}\). The imaginary part \(\pm q\) comes from \(\dfrac{\sqrt{4ac - b^2}}{2a}\).
  • If one root is known, the other is its conjugate. The sum of the roots is \(2p = -\dfrac{b}{a}\) and their product is \(p^2 + q^2 = \dfrac{c}{a}\).

Plotting the roots

  • Plot \(p + qi\) as the point \((p, q)\): real part across, imaginary part up.
  • Label both points and the axes: Re and Im.

On the calculator

Graphing a quadratic and reading the vertex

  • Graph the function and find its lowest point (a minimum) or highest point (a maximum) with the graph tools.
  • If the vertex is above the x-axis on an upward parabola, or below it on a downward one, there are no real roots.
  • Write the vertex as a coordinate pair, for example \((-2, 3)\).

Screenshot and video to come

Solving in complex mode

  • Check that the calculator is in Complex mode (Rectangular) in the document settings.
  • Use the polynomial or quadratic solver, or solve the equation, and read both roots.
  • Write the two roots \(p \pm qi\) and check they are a conjugate pair.

Screenshot and video to come

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
A quadratic with no real roots

Use technology for this question. (a) Graph \(y = 2x^2 + 8x + 11\) and write down the coordinates of the vertex. (b) Solve \(2x^2 + 8x + 11 = 0\). (c) Show the solutions on an Argand diagram.

Working
  1. −6−5−4−3−2−11224681012xyyvertex (-2, 3)
    Graph

    From the calculator (graph): vertex \((-2, 3)\)

    The parabola opens upwards and its lowest point is above the x-axis, so it never crosses it. There are no real roots.

  2. Solve in complex mode

    Complex mode: \(x = -2 \pm 1.22i\)

    Exact value: \(x = -2 \pm \frac{\sqrt{6}}{2}i\). The calculator must be in complex mode, or it reports no solution.

  3. −4−3−2−112−2−112ReImPQ
    Plot the roots

    \(P(-2, 1.22)\) and \(Q(-2, -1.22)\)

    The real part is across and the imaginary part is up. The two points are reflections of each other in the real axis: a conjugate pair. Their real part, \(-2\), is the x-coordinate of the vertex.

Video walkthrough coming
Roots as a conjugate pair

Solve \(0.5x^2 - 3x + 7 = 0\) with the calculator in complex mode, and show the roots on an Argand diagram.

Working
  1. Check the discriminant

    \(b^2 - 4ac = (-3)^2 - 4(0.5)(7) = -5\)

    Negative, so there are no real roots and the solver must be in complex mode.

  2. Solve

    Complex mode: \(x = 3 \pm 2.24i\)

    Exact: \(x = 3 \pm \sqrt{5}\, i\). The two roots are conjugates.

  3. −1123456−3−2−1123ReImPQ
    Plot

    \((3, 2.24)\) and \((3, -2.24)\)

    Same real part, opposite imaginary parts: the points are mirror images in the real axis.

Video walkthrough coming