Question

write the given quotient in the form $a + bi$.
$\frac{6 - 4i}{2 + 7i}$
$\frac{6 - 4i}{2 + 7i} = \square$
(simplify your answer. use integers or fractions for any numbers in the

Explanation:

Step1: Multiply by conjugate of denominator

Multiply numerator and denominator by $2-7i$ (the conjugate of $2+7i$):
$$\frac{6-4i}{2+7i} \times \frac{2-7i}{2-7i} = \frac{(6-4i)(2-7i)}{(2+7i)(2-7i)}$$

Step2: Expand numerator using FOIL

Calculate product of numerator terms:
$$(6)(2) + (6)(-7i) + (-4i)(2) + (-4i)(-7i) = 12 - 42i - 8i + 28i^2$$
Substitute $i^2=-1$:
$$12 - 50i + 28(-1) = 12 - 28 - 50i = -16 - 50i$$

Step3: Expand denominator (difference of squares)

Calculate product of denominator terms:
$$2^2 - (7i)^2 = 4 - 49i^2$$
Substitute $i^2=-1$:
$$4 - 49(-1) = 4 + 49 = 53$$

Step4: Split into $a+bi$ form

Divide real and imaginary parts by denominator:
$$\frac{-16 - 50i}{53} = \frac{-16}{53} - \frac{50}{53}i$$

Answer:

$\frac{-16}{53} - \frac{50}{53}i$