Complex roots of quadratic equations Problem set B Worked solutions

Use technology for this question. Consider the quadratic \(y = 3x^2 + 6x + 5\).

1. (a) Graph the function and write down the coordinates of the vertex.

   • From the calculator (graph): vertex \((-1, 2)\)
   • The vertex is above the x-axis, so there are no real roots.

2. (b) Solve \(3x^2 + 6x + 5 = 0\) with the calculator in complex mode.

   • Discriminant: \(b^2 - 4ac = -24\)
   • Complex mode: \(x = -1 \pm 0.816i\)

3. (c) Show the solutions on an Argand diagram and write down the coordinates of the two points. Compare the real part of the roots with the x-coordinate of the vertex.

   • The points are \((-1, 0.816)\) and \((-1, -0.816)\).
   • They are a conjugate pair, reflections in the real axis. The real part \(-1\) is the x-coordinate of the vertex.

Use the calculator in complex mode to solve each equation. Give exact values where you can and otherwise 3 significant figures.

1. (a) \(x^2 + 64 = 0\)

   • \(b^2 - 4ac = -256 < 0\), so the roots are complex.
   • Complex mode: \(x = \pm 8i\)

2. (b) \(x^2 + 8x + 20 = 0\)

   • \(b^2 - 4ac = -16 < 0\), so the roots are complex.
   • Complex mode: \(x = -4 \pm 2i\)

3. (c) \(2x^2 + 3x + 5 = 0\)

   • \(b^2 - 4ac = -31 < 0\), so the roots are complex.
   • Complex mode: \(x = -0.75 \pm 1.39i\)

One root of a quadratic equation with real coefficients is \(4 - 3i\).

1. (a) Write down the other root.

   • Complex roots of a real quadratic come as a conjugate pair.
   • The other root is \(4 + 3i\).

2. (b) Show both roots on an Argand diagram.

   • The points are \((4, -3)\) and \((4, 3)\).
   • They are reflections of each other in the real axis.

3. (c) Find the distance between the two points.

   • The points are 3 above and 3 below the real axis.
   • Distance \(= 2 \times 3 = 6\)

The distance \(d\) metres between a drone and a buoy, \(t\) seconds after take-off, is modelled by \(d = 0.5t^2 - 4t + 11\). Use technology for this question.

1. (a) Graph \(d\) and write down the coordinates of the vertex. What does the vertex tell you about the drone?

   • From the calculator (graph): vertex \((4, 3)\)
   • The parabola opens upwards and its lowest point is above the t-axis, so \(d\) is always positive.
   • The smallest distance is 3 m, at \(t = 4\).

2. (b) Solve \(d = 0\) with the calculator in complex mode.

   • Complex mode: \(0.5t^2 - 4t + 11 = 0\) gives \(t = 4 \pm 2.45i\)
   • Exact: \(t = 4 \pm \sqrt{6}\,i\)

3. (c) What does the answer to part (b) tell you about the drone reaching the buoy?

   • Reaching the buoy means a real \(t\) with \(d = 0\).
   • The solutions are not real, so the drone never reaches it. This matches the graph never reaching the t-axis.

A quadratic equation \(x^2 + bx + c = 0\) has real coefficients \(b\) and \(c\). One root is \(-3 + 4i\).

1. (a) Write down the other root.

   • The coefficients are real, so the roots are a conjugate pair.
   • The other root is \(-3 - 4i\).

2. (b) Find \(b\) and \(c\).

   • Sum of roots \(= -6 = -b\), so \(b = 6\).
   • Product of roots \(= (-3 + 4i)(-3 - 4i) = (-3)^2 + 4^2 = 25 = c\).
   • So the equation is \(x^2 + 6x + 25 = 0\).

Use technology for this question. Consider \(x^2 + 8x + k = 0\), where \(k\) is a real number.

1. (a) Write down the x-coordinate of the vertex of \(y = x^2 + 8x + k\).

   • The vertex is at \(x = -\dfrac{b}{2a} = -4\).

2. (b) Find the values of \(k\) for which the equation has no real roots.

   • The vertex has y-coordinate \(k - 16\).
   • The parabola opens upwards, so there are no real roots when the vertex is above the axis: \(k - 16 > 0\), \(k > 16\).

3. (c) Take \(k = 41\). Solve the equation, and find the distance between the two roots on an Argand diagram.

   • \(k = 41 > 16\), so the roots are complex.
   • Complex mode: \(x = -4 \pm 5i\)
   • The points are 5 above and 5 below the real axis, so the distance is \(2 \times 5 = 10\).