Question

express as a complex number in simplest a+bi form: \\(\frac{-26 - 26i}{-6 + 4i}\\) answer attempt 1 out of 2

Explanation:

Step1: Factor numerator and denominator

Numerator: $-26 - 26i = -26(1+i)$
Denominator: $-6 + 4i = 2(-3 + 2i)$
Simplify fraction: $\frac{-26(1+i)}{2(-3 + 2i)} = \frac{-13(1+i)}{-3 + 2i}$

Step2: Multiply by conjugate of denominator

Conjugate of $-3+2i$ is $-3-2i$.
$\frac{-13(1+i)(-3-2i)}{(-3+2i)(-3-2i)}$

Step3: Expand numerator

$(-13)[(1)(-3)+(1)(-2i)+i(-3)+i(-2i)] = (-13)[-3-2i-3i-2i^2]$
Since $i^2=-1$: $(-13)[-3-5i-2(-1)] = (-13)[-3-5i+2] = (-13)(-1-5i) = 13+65i$

Step4: Expand denominator

$(-3)^2 - (2i)^2 = 9 - 4i^2 = 9 - 4(-1) = 9+4=13$

Step5: Divide numerator by denominator

$\frac{13+65i}{13} = 1 + 5i$

Answer:

$1 + 5i$