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
To find the values of a and b from the given equation a + bi = (1+i)/(1-2i), we need to simplify the complex fraction on the right-hand side. Step 1: Multiply the numerator and denominator by the conjugate of the denominator. The conjugate of 1-2i is 1+2i. a + bi = (1+i)/(1-2i) × (1+2i)/(1+2i) Step 2: Perform the multiplication in the numerator. (1+i)(1+2i) = 1(1) + 1(2i) + i(1) + i(2i) = 1 + 2i + i + 2i^2 Since i^2 = -1: = 1 + 3i + 2(-1) = 1 + 3i - 2 = -1 + 3i Step 3: Perform the multiplication in the denominator. (1-2i)(1+2i) This is in the form (x-y)(x+y) = x^2 - y^2. = 1^2 - (2i)^2 = 1 - 4i^2 Since i^2 = -1: = 1 - 4(-1) = 1 + 4 = 5 Step 4: Write the simplified complex number in the form a+bi. a + bi = (-1 + 3i)/(5) a + bi = -(1)/(5) + (3)/(5)i Step 5: Identify the values of a and b. By comparing the real and imaginary parts, we get: a = -(1)/(5) b = (3)/(5) The correct option is C. What's next?