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

Question

Working without a calculator, find the roots of z^2 - (1 + 2i)z + 1 + 7i = 0, giving each in cartesian form a + ib. [6]

The equation does not factorise by inspection, so use the quadratic formula. It is valid over the complex numbers, but the square root of a complex number then has to be found by hand. Step 1: apply the formula. For z² - (1 + 2i)z + (1 + 7i) = 0 the coefficients are a = 1, b = -(1 + 2i), c = 1 + 7i. z = [ -b +/- sqrt(b² - 4ac) ]/(2a) = [ (1 + 2i) +/- sqrt( (1 + 2i)² - 4(1 + 7i) ) ]/2 Step 2: simplify the discriminant. (1 + 2i)² = 1 + 2(1)(2i) + (2i)² = 1 + 4i + 4i² = 1 + 4i - 4 = -3 + 4i -4(1 + 7i) = -4 - 28i b² - 4ac = (-3 + 4i) + (-4 - 28i) = -7 - 24i Step 3: find sqrt(-7 - 24i). Let it be u + vi with u and v real. (u + vi)² = u² + 2uvi + v² i² = (u² - v²) + 2uv i Set this equal to -7 - 24i and compare real and imaginary parts: real: u² - v² = -7 ...(1) imaginary: 2uv = -24, so uv = -12 and v = -12/u ...(2) Substitute (2) into (1): u² - (-12/u)² = -7 u² - 144/u² = -7 Multiply through by u² (u is not 0, since uv = -12): u⁴ + 7u² - 144 = 0 This is a quadratic in u². Since 16 x (-9) = -144 and 16 - 9 = 7: (u² + 16)(u² - 9) = 0 u is real, so u² cannot be -16; hence u² = 9 and u = 3 or u = -3. u = 3 gives v = -12/3 = -4 (and u = -3 gives v = 4, the negative of the same root). So sqrt(-7 - 24i) = +/- (3 - 4i). Step 4: substitute back into the formula. z = [ (1 + 2i) + (3 - 4i) ]/2 = (4 - 2i)/2 = 2 - i or z = [ (1 + 2i) - (3 - 4i) ]/2 = (-2 + 6i)/2 = -1 + 3i Check: sum of roots = (2 - i) + (-1 + 3i) = 1 + 2i, and product = (2 - i)(-1 + 3i) = -2 + 6i + i - 3i² = -2 + 7i + 3 = 1 + 7i, which match the coefficients of the equation. So z = 2 - i or z = -1 + 3i.