Complex Numbers and Quadratic Equations

13

Step 1: Notice that (2 + 3i)(2 - 3i) is of the form (a + ib)(a - ib) = a² + b². Step 2: Here a = 2 and b = 3. Step 3: (2)² + (3)² = 4 + 9 = 13. Final Answer: 13 (a real number). This also equals |z|² where z = 2 + 3i. A common mistake is writing 4 - 9i² = 4 - (-9) but forgetting to simplify to 13.

z · z̄

Step 1: We know |z|² = a² + b² for z = a + ib. Step 2: Compute z · z̄ = (a + ib)(a - ib) = a² - (ib)² = a² - i²b² = a² + b². Step 3: So z · z̄ = a² + b² = |z|². Final Answer: z · z̄. Note that z + z̄ = 2a (twice the real part), not |z|².

i

Step 1: Powers of i repeat in a cycle of 4: i¹=i, i²=-1, i³=-i, i⁴=1, then it repeats. Step 2: Divide the exponent 97 by 4 to find the remainder: 97 ÷ 4 = 24 remainder 1. Step 3: So i⁹⁷ = i^(4×24 + 1) = (i⁴)²⁴ × i¹ = 1²⁴ × i = i. Final Answer: i. A common mistake is dividing by 2 instead of 4 when finding the remainder.

6

Step 1: Any complex number is written in the form z = a + ib, where a is the real part and b is the imaginary part. Step 2: Here z = 6 - 9i = 6 + (-9)i, so a = 6 and b = -9. Step 3: Re(z) = a = 6. Final Answer: 6. A common mistake is confusing the real part with the imaginary part.