Complex roots of quadratic equations Problem set A

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

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

1. (a) \(x^2 + 49 = 0\)
2. (b) \(x^2 - 6x + 25 = 0\)
3. (c) \(3x^2 + 2x + 4 = 0\)

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

1. (a) Write down the other root.
2. (b) Show both roots on an Argand diagram.
3. (c) Find the distance between the two points.

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?
2. (b) Solve \(P = 0\) with the calculator in complex mode.
3. (c) What does the answer to part (b) tell you about break-even prices?

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

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 \(y = x^2 - 6x + k\).
2. (b) Find the values of \(k\) for which the equation has no real roots.
3. (c) Take \(k = 18\). Solve the equation, and find the distance between the two roots on an Argand diagram.