If (2 +4i)² -2(x +iy) =x +iy Find x and y
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: Expand the term $(2+4i)^2$.
Using the formula $(a+b)^2 = a^2 + 2ab + b^2$:
$$ (2+4i)^2 = 2^2 + 2(2)(4i) + (4i)^2 $$
$$ = 4 + 16i + 16i^2 $$
Since $i^2 = -1$:
$$ = 4 + 16i - 16 $$
$$ = -12 + 16i $$
Step 2: Substitute this back into the original equation.
The equation is $(2 +4i)^2 -2(x +iy) =x +iy$.
$$ (-12 + 16i) - 2(x + iy) = x + iy $$
Step 3: Distribute the $-2$ on the left side.
$$ -12 + 16i - 2x - 2iy = x + iy $$
Step 4: Group the real and imaginary parts on the left side of the equation.
$$ (-12 - 2x) + (16 - 2y)i = x + iy $$
Step 5: Equate the real parts and the imaginary parts from both sides of the equation.
For the real parts:
$$ -12 - 2x = x $$
For the imaginary parts:
$$ 16 - 2y = y $$
Step 6: Solve the system of equations for $x$ and $y$.
From the real parts equation:
$$ -12 = x + 2x $$
$$ -12 = 3x $$
$$ x = \frac{-12}{3} $$
$$ x = -4 $$
From the imaginary parts equation:
$$ 16 = y + 2y $$
$$ 16 = 3y $$
$$ y = \frac{16}{3} $$
The values are $x=-4$ and $y=\frac{16}{3}$.
The final answer is $\boxed{x=-4, y=\frac{16}{3}}$.
