This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Identify the given complex numbers. We are given two complex numbers: z_1 = 2+3i z_2 = -3+i Step 2: Calculate the product z_1 z_2. Multiply the two complex numbers: z_1 z_2 = (2+3i)(-3+i) Use the distributive property (FOIL method): z_1 z_2 = 2(-3) + 2(i) + 3i(-3) + 3i(i) z_1 z_2 = -6 + 2i - 9i + 3i^2 Step 3: Substitute i^2 = -1. Recall that i^2 = -1. Substitute this into the expression: z_1 z_2 = -6 + 2i - 9i + 3(-1) z_1 z_2 = -6 + 2i - 9i - 3 Step 4: Combine the real and imaginary parts. Group the real terms and the imaginary terms: z_1 z_2 = (-6 - 3) + (2i - 9i) z_1 z_2 = -9 - 7i Step 5: Identify the real part of z_1 z_2. The real part of a complex number a+bi is a. In this case, the real part of -9-7i is -9. The final answer is -9.