Complex roots of quadratic equations Problem set A Worked solutions

Use technology for this question. Consider the quadratic \(y = x^2 - 4x + 7\).

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

   • From the calculator (graph): vertex \((2, 3)\)
   • The vertex is above the x-axis, so there are no real roots.

2. (b) Solve \(x^2 - 4x + 7 = 0\) with the calculator in complex mode.

   • Discriminant: \(b^2 - 4ac = -12\)
   • Complex mode: \(x = 2 \pm 1.73i\)

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 \((2, 1.73)\) and \((2, -1.73)\).
   • They are a conjugate pair, reflections in the real axis. The real part \(2\) 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) \(x^2 + 49 = 0\)

   • \(b^2 - 4ac = -196 < 0\), so the roots are complex.
   • Complex mode: \(x = \pm 7i\)

2. (b) \(x^2 - 6x + 25 = 0\)

   • \(b^2 - 4ac = -64 < 0\), so the roots are complex.
   • Complex mode: \(x = 3 \pm 4i\)

3. (c) \(3x^2 + 2x + 4 = 0\)

   • \(b^2 - 4ac = -44 < 0\), so the roots are complex.
   • Complex mode: \(x = -0.333 \pm 1.11i\)

One root of a quadratic equation with real coefficients is \(2 + 5i\).

1. (a) Write down the other root.

   • Complex roots of a real quadratic come as a conjugate pair.
   • The other root is \(2 - 5i\).

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

   • The points are \((2, 5)\) and \((2, -5)\).
   • They are reflections of each other in the real axis.

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

   • The points are 5 above and 5 below the real axis.
   • Distance \(= 2 \times 5 = 10\)

The weekly profit \(P\) (in thousands of euros) of a stall that charges \(x\) euros for an item is modelled by \(P = -2x^2 + 12x - 21\). Use technology for this question.

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

   • From the calculator (graph): vertex \((3, -3)\)
   • The parabola opens downwards and its highest point is below the x-axis, so \(P\) is always negative.
   • The greatest profit is \(-3\) thousand euros, a loss, at \(x = 3\).

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

   • Complex mode: \(-2x^2 + 12x - 21 = 0\) gives \(x = 3 \pm 1.22i\)
   • Exact: \(x = 3 \pm \dfrac{\sqrt{6}}{2}\,i\)

3. (c) What does the answer to part (b) tell you about break-even prices?

   • A break-even price is a real \(x\) with \(P = 0\).
   • The solutions are not real, so there is none. This matches the graph never reaching the x-axis.

A quadratic equation \(x^2 + bx + c = 0\) has real coefficients \(b\) and \(c\). One root is \(2 + 5i\).

1. (a) Write down the other root.

   • The coefficients are real, so the roots are a conjugate pair.
   • The other root is \(2 - 5i\).

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

   • Sum of roots \(= 4 = -b\), so \(b = -4\).
   • Product of roots \(= (2 + 5i)(2 - 5i) = (2)^2 + 5^2 = 29 = c\).
   • So the equation is \(x^2 - 4x + 29 = 0\).

Use technology for this question. Consider \(x^2 - 6x + k = 0\), where \(k\) is a real number.

1. (a) Write down the x-coordinate of the vertex of \(y = x^2 - 6x + k\).

   • The vertex is at \(x = -\dfrac{b}{2a} = 3\).

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

   • The vertex has y-coordinate \(k - 9\).
   • The parabola opens upwards, so there are no real roots when the vertex is above the axis: \(k - 9 > 0\), \(k > 9\).

3. (c) Take \(k = 18\). Solve the equation, and find the distance between the two roots on an Argand diagram.

   • \(k = 18 > 9\), so the roots are complex.
   • Complex mode: \(x = 3 \pm 3i\)
   • The points are 3 above and 3 below the real axis, so the distance is \(2 \times 3 = 6\).