A-Level Complex numbers in cartesian form and the Argand diagram: worked solution

Question

The three roots of z^3 + az^2 + bz - 16sqrt2 = 0 (a = 4 - 2sqrt2, b = 8 - 8sqrt2) are A (-2 + 2i), B and C (real root 2sqrt2). Plot them on an Argand diagram, giving each modulus and argument, and state a geometric link between A and B.

First find B, which is not given. For a cubic z³ + az² + bz + (constant) = 0, the sum of the three roots equals -a. Sum of roots = -(4 - 2sqrt2) = -4 + 2sqrt2 Two roots are known, A = -2 + 2i and C = 2sqrt2, so B = (-4 + 2sqrt2) - (-2 + 2i) - 2sqrt2 B = -4 + 2sqrt2 + 2 - 2i - 2sqrt2 B = -2 - 2i Now the modulus of each root, using |x + iy| = sqrt(x² + y²). |A| = |-2 + 2i| = sqrt((-2)² + 2²) = sqrt(4 + 4) = sqrt8 = 2sqrt2 |B| = |-2 - 2i| = sqrt((-2)² + (-2)²) = sqrt8 = 2sqrt2 |C| = |2sqrt2| = 2sqrt2 Now the arguments, taken in the range (-pi, pi]. A = -2 + 2i has a negative real part and a positive imaginary part, so it lies in the SECOND quadrant. The acute angle it makes with the real axis is arctan(2/2) = arctan 1 = pi/4, measured from the NEGATIVE real axis, so arg A = pi - pi/4 = 3pi/4 B = -2 - 2i lies in the THIRD quadrant, the same distances but below the real axis, so the argument has the same size and the opposite sign: arg B = -(pi - pi/4) = -3pi/4 C = 2sqrt2 is a positive real number, lying on the positive real axis, so arg C = 0 Argand diagram: all three points lie on the circle centred at the origin with radius 2sqrt2. Plot A at (-2, 2) (up and to the left), B at (-2, -2) (directly below A), and C at (2sqrt2, 0) = (2.83, 0) on the positive real axis. Geometric link: B = -2 - 2i is the complex conjugate of A = -2 + 2i (same real part, imaginary part negated), so B is the REFLECTION of A in the real axis.