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: Isolate z by multiplying both sides by (z+2). We are given the equation: (z)/(z+2) = 2-i Multiply both sides by (z+2): z = (2-i)(z+2) Step 2: Expand the right side of the equation. z = 2(z) + 2(2) - i(z) - i(2) z = 2z + 4 - iz - 2i Step 3: Gather all terms containing z on one side and constant terms on the other side. Subtract 2z and add iz to both sides: z - 2z + iz = 4 - 2i -z + iz = 4 - 2i Step 4: Factor out z from the terms on the left side. z(-1 + i) = 4 - 2i Step 5: Solve for z by dividing both sides by (-1+i). z = (4 - 2i)/(-1 + i) Step 6: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of -1+i is -1-i. z = (4 - 2i)/(-1 + i) × (-1 - i)/(-1 - i) Step 7: Perform the multiplication in the numerator. (4 - 2i)(-1 - i) = 4(-1) + 4(-i) - 2i(-1) - 2i(-i) = -4 - 4i + 2i + 2i^2 Since i^2 = -1: = -4 - 2i + 2(-1) = -4 - 2i - 2 = -6 - 2i Step 8: Perform the multiplication in the denominator. (-1 + i)(-1 - i) = (-1)^2 - (i)^2 = 1 - (-1) = 1 + 1 = 2 Step 9: Combine the simplified numerator and denominator. z = (-6 - 2i)/(2) Step 10: Simplify the expression to the form a+bi. z = (-6)/(2) - (2i)/(2) z = -3 - i The value of z in the form a+bi is -3 - i. Last free one today — make it count tomorrow, or type /upgrade for unlimited.