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: Write the expression for (z_1)/(z_2). Given z_1 = 3+2i and z_2 = -1-5i, we have: (z_1)/(z_2) = (3+2i)/(-1-5i) Step 2: Multiply the numerator and denominator by the conjugate of the denominator. The conjugate of -1-5i is -1+5i. (3+2i)/(-1-5i) × (-1+5i)/(-1+5i) Step 3: Expand the numerator. (3+2i)(-1+5i) = 3(-1) + 3(5i) + 2i(-1) + 2i(5i) = -3 + 15i - 2i + 10i^2 Since i^2 = -1: = -3 + 13i + 10(-1) = -3 + 13i - 10 = -13 + 13i Step 4: Expand the denominator. The denominator is in the form (a-bi)(a+bi) = a^2 + b^2. Here, a=-1 and b=5. (-1-5i)(-1+5i) = (-1)^2 + (5)^2 = 1 + 25 = 26 Step 5: Combine the simplified numerator and denominator. (z_1)/(z_2) = (-13+13i)/(26) Step 6: Simplify the expression by dividing both the real and imaginary parts by the denominator. (z_1)/(z_2) = (-13)/(26) + (13)/(26)i (z_1)/(z_2) = -(1)/(2) + (1)/(2)i Step 7: Compare the result with the given options. The calculated answer is -(1)/(2) + (1)/(2)i. Based on the image, option A appears to be -(1)/(2) + (1)/(2)i. Due to the blurriness of the image, this is the closest match. The final answer is A. -(1)/(2) + (1)/(2)i.