Question

express as a complex number in simplest $a+bi$ form: $\frac{-27 + 21i}{1 + 5i}$
answer attempt 2 out of 3

Explanation:

Step1: Multiply by conjugate of denominator

Multiply numerator and denominator by $1-5i$:
$$\frac{-27+21i}{1+5i} \times \frac{1-5i}{1-5i} = \frac{(-27+21i)(1-5i)}{(1+5i)(1-5i)}$$

Step2: Expand numerator using FOIL

Calculate product for numerator:
$$(-27)(1) + (-27)(-5i) + 21i(1) + 21i(-5i) = -27 + 135i + 21i - 105i^2$$
Substitute $i^2=-1$:
$$-27 + 156i - 105(-1) = -27 + 156i + 105 = 78 + 156i$$

Step3: Expand denominator (difference of squares)

Calculate product for denominator:
$$1^2 - (5i)^2 = 1 - 25i^2$$
Substitute $i^2=-1$:
$$1 - 25(-1) = 1 + 25 = 26$$

Step4: Simplify the fraction

Divide each term by 26:
$$\frac{78}{26} + \frac{156}{26}i = 3 + 6i$$

Answer:

$3 + 6i$