Question

question
express as a complex number in simplest a+bi form:
\frac{22 + 24i}{-5 + 9i}
answer attempt 1 out of 2

Explanation:

Step1: Multiply by conjugate of denominator

Multiply numerator and denominator by $-5-9i$ (the conjugate of $-5+9i$):
$$\frac{22+24i}{-5+9i} \times \frac{-5-9i}{-5-9i} = \frac{(22+24i)(-5-9i)}{(-5+9i)(-5-9i)}$$

Step2: Expand numerator using FOIL

Calculate the product of the numerator:
$$(22)(-5) + (22)(-9i) + (24i)(-5) + (24i)(-9i) = -110 - 198i - 120i - 216i^2$$
Substitute $i^2=-1$:
$$-110 - 318i - 216(-1) = -110 - 318i + 216 = 106 - 318i$$

Step3: Expand denominator

Calculate the product of the denominator (difference of squares):
$$(-5)^2 - (9i)^2 = 25 - 81i^2$$
Substitute $i^2=-1$:
$$25 - 81(-1) = 25 + 81 = 106$$

Step4: Simplify the fraction

Divide numerator by denominator:
$$\frac{106 - 318i}{106} = \frac{106}{106} - \frac{318}{106}i$$

Step5: Reduce the coefficients

Simplify the numerical terms:
$$1 - 3i$$

Answer:

$1 - 3i$