Question

express as a complex number in simplest a+bi form: $\frac{7 + 3i}{-2 + 6i}$

Explanation:

Step1: Multiply by conjugate of denominator

Multiply numerator and denominator by $-2-6i$:
$$\frac{7+3i}{-2+6i} \times \frac{-2-6i}{-2-6i} = \frac{(7+3i)(-2-6i)}{(-2+6i)(-2-6i)}$$

Step2: Expand numerator using FOIL

Calculate product of the numerator:
$$(7)(-2) + (7)(-6i) + (3i)(-2) + (3i)(-6i) = -14 -42i -6i -18i^2$$
Substitute $i^2=-1$:
$$-14 -48i -18(-1) = -14 -48i +18 = 4 -48i$$

Step3: Expand denominator (difference of squares)

Calculate product of the denominator:
$$(-2)^2 - (6i)^2 = 4 - 36i^2$$
Substitute $i^2=-1$:
$$4 - 36(-1) = 4 + 36 = 40$$

Step4: Simplify the fraction

Divide numerator by denominator:
$$\frac{4 -48i}{40} = \frac{4}{40} - \frac{48}{40}i = \frac{1}{10} - \frac{6}{5}i$$

Answer:

$\frac{1}{10} - \frac{6}{5}i$