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

Step 1: use the conjugate root theorem. The coefficients 1, a, b and -16sqrt2 are all real, so any non-real roots come in conjugate pairs. Since -2 + 2i is a root, its conjugate -2 - 2i is also a root. Step 2: build the quadratic factor those two roots produce. Sum of the pair = (-2 + 2i) + (-2 - 2i) = -4 Product of the pair = (-2 + 2i)(-2 - 2i) = (-2)² - (2i)² = 4 - (-4) = 8 A quadratic with this sum and product is z² - (sum)z + (product): z² - (-4)z + 8 = z² + 4z + 8 So z² + 4z + 8 is a factor of the cubic. Step 3: find the third root. The cubic is degree 3 with leading coefficient 1, so it factorises as z³ + az² + bz - 16sqrt2 = (z² + 4z + 8)(z - c) where c is the remaining root. Compare the constant terms on each side: 8 × (-c) = -16sqrt2 -8c = -16sqrt2 c = 2sqrt2 Step 4: expand the factorisation to read off a and b. (z² + 4z + 8)(z - 2sqrt2) = z³ + 4z² + 8z - 2sqrt2 z² - 8sqrt2 z - 16sqrt2 = z³ + (4 - 2sqrt2)z² + (8 - 8sqrt2)z - 16sqrt2 Comparing with z³ + az² + bz - 16sqrt2: coefficient of z²: a = 4 - 2sqrt2 coefficient of z: b = 8 - 8sqrt2 Step 5: state the roots. The three roots are z = -2 + 2i, z = -2 - 2i and z = 2sqrt2, so the two remaining roots are -2 - 2i and 2sqrt2.

Step 1: expand the product before substituting, which keeps the algebra light. (z - 2)(z + 1) = z² + z - 2z - 2 = z² - z - 2 Step 2: work out z² for z = x + 2i. z² = (x + 2i)² = x² + 2(x)(2i) + (2i)² (2i)² = 4i² = -4, since i² = -1. So z² = x² - 4 + 4xi Step 3: substitute into the expanded expression. z² - z - 2 = (x² - 4 + 4xi) - (x + 2i) - 2 Collect the real terms: x² - 4 - x - 2 = x² - x - 6 Collect the imaginary terms: 4xi - 2i = (4x - 2)i So (z - 2)(z + 1) = (x² - x - 6) + (4x - 2)i Step 4: use the condition that the answer is mi. mi is purely imaginary with m real, so its real part is 0 and its imaginary part is m. Two complex numbers are equal only if their real parts are equal AND their imaginary parts are equal, so compare them separately. Real parts: x² - x - 6 = 0 Imaginary parts: 4x - 2 = m Step 5: solve the real-part equation. x² - x - 6 = 0 Look for two numbers multiplying to -6 and adding to -1: those are -3 and +2. (x - 3)(x + 2) = 0 So x = 3 or x = -2. The question states that x is a positive real number, so reject x = -2. x = 3 Step 6: find m. m = 4x - 2 = 4(3) - 2 = 12 - 2 = 10 Check: z = 3 + 2i, so z - 2 = 1 + 2i and z + 1 = 4 + 2i, and (1 + 2i)(4 + 2i) = 4 + 2i + 8i + 4i² = 4 + 10i - 4 = 10i. This is mi with m = 10, as required. Step 7: find the complex number represented by B. With x = 3 we have z = 3 + 2i, so z - 2 = 1 + 2i. B represents the product z(z - 2): z(z - 2) = (3 + 2i)(1 + 2i) Multiply out term by term: 3(1) + 3(2i) + 2i(1) + 2i(2i) = 3 + 6i + 2i + 4i² = 3 + 8i - 4 (using i² = -1) = -1 + 8i So B represents -1 + 8i.

The two equations are (1 + i)z + 2w = -2 + 4i ... (1) 3z - w = 4 + 2i ... (2) Treat z and w as unknowns and eliminate one of them exactly as with real simultaneous equations. Step 1: make w the subject of (2), because its coefficient is simplest. w = 3z - 4 - 2i. Step 2: substitute into (1). (1 + i)z + 2(3z - 4 - 2i) = -2 + 4i (1 + i)z + 6z - 8 - 4i = -2 + 4i Step 3: collect the z terms on the left and the constants on the right. (1 + i + 6)z = -2 + 4i + 8 + 4i (7 + i)z = 6 + 8i Step 4: divide, and remove i from the denominator by multiplying top and bottom by the conjugate of the denominator, 7 - i. z = (6 + 8i)/(7 + i) = (6 + 8i)(7 - i) / [(7 + i)(7 - i)] Denominator: (7 + i)(7 - i) = 49 - i² = 49 + 1 = 50. Numerator: (6)(7) + (6)(-i) + (8i)(7) + (8i)(-i) = 42 - 6i + 56i - 8i² = 42 + 50i + 8 = 50 + 50i. So z = (50 + 50i)/50 = 1 + i. Step 5: substitute back into the expression from Step 1 to get w. w = 3(1 + i) - 4 - 2i = 3 + 3i - 4 - 2i = -1 + i. Check in (1): (1 + i)(1 + i) + 2(-1 + i) = (1 + 2i + i²) + (-2 + 2i) = 2i - 2 + 2i = -2 + 4i. Correct.