(2-3i)z+(4+i)z=-(1+3i)^2

3 min read Jun 16, 2024

Solving Complex Equations: (2-3i)z + (4+i)z = -(1+3i)^2

This article will guide you through the steps of solving the complex equation: (2-3i)z + (4+i)z = -(1+3i)^2. We will leverage the properties of complex numbers to simplify the equation and isolate the variable 'z'.

1. Simplifying the Equation

First, let's simplify the equation by combining like terms on the left-hand side and expanding the right-hand side.

Combining Like Terms: (2-3i)z + (4+i)z = (2+4)z + (-3+1)i*z = 6z - 2iz
Expanding the Right-Hand Side: -(1+3i)^2 = - (1 + 6i + 9i^2) = - (1 + 6i - 9) = 8 - 6i

Now, our equation looks like this: 6z - 2iz = 8 - 6i.

2. Isolating 'z'

To isolate 'z', we need to express it as a single complex number. This can be done by factoring out 'z' on the left-hand side and then dividing both sides by the coefficient of 'z':

Factoring out 'z': z(6 - 2i) = 8 - 6i
Dividing both sides by (6 - 2i): z = (8 - 6i) / (6 - 2i)

3. Rationalizing the Denominator

The denominator is a complex number. To simplify further, we need to rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator:

Conjugate of (6 - 2i) is (6 + 2i) z = (8 - 6i) / (6 - 2i) * (6 + 2i) / (6 + 2i)
Expanding both numerator and denominator: z = (48 + 16i - 36i - 12i^2) / (36 - 4i^2)
Simplifying (remember i^2 = -1): z = (48 + 16i - 36i + 12) / (36 + 4) = (60 - 20i) / 40

4. Final Solution

Finally, we can simplify the complex number by dividing both the real and imaginary parts by 40:

z = (60/40) - (20/40)i = 3/2 - 1/2i

Therefore, the solution to the equation (2-3i)z + (4+i)z = -(1+3i)^2 is z = 3/2 - 1/2i.