Question

question
express as a complex number in simplest a+bi form:
\frac{-7 + 3i}{5 + 2i}
answer attempt 1 out of 2

Explanation:

Step1: Multiply by conjugate of denominator

Multiply numerator and denominator by $5-2i$:
$$\frac{-7+3i}{5+2i} \times \frac{5-2i}{5-2i} = \frac{(-7+3i)(5-2i)}{(5+2i)(5-2i)}$$

Step2: Expand numerator using FOIL

Calculate product of numerator terms:
$$(-7)(5) + (-7)(-2i) + 3i(5) + 3i(-2i) = -35 + 14i + 15i -6i^2$$
Substitute $i^2=-1$:
$$-35 + 29i -6(-1) = -35 + 29i +6 = -29 +29i$$

Step3: Expand denominator (difference of squares)

Calculate denominator product:
$$5^2 - (2i)^2 = 25 -4i^2$$
Substitute $i^2=-1$:
$$25 -4(-1) = 25 +4 =29$$

Step4: Simplify the fraction

Divide numerator by denominator:
$$\frac{-29 +29i}{29} = \frac{-29}{29} + \frac{29i}{29}$$

Answer:

$-1 + i$