Pre-Calculus - Find the Quotient of Complex Numbers

Simplify $\frac{3+5i}{2-4i}$ into a number of the form .

$-\frac{7}{10}+\frac{11}{10}i$

$\frac{1}{2}-\frac{3}{2}i$

None of the other answers.

Explanation

We have

$\frac{3+5i}{2-4i}$

Multiply by the complex conjugate of the denominator.

The complex conjugate is the denominator with the sign changed:

$\frac{3+5i}{2-4i}\times \frac{2+4i}{2+4i}$

Multiply fractions

$\frac{(3+5i)(2+4i)}{(2-4i)(2+4i)}$

FOIL the numerator and denominator

$\frac{6+12i+10i+20i^2}{4+8i-8i-16i^2}$

Apply the rule of :

$\frac{6+12i+10i-20}{4+8i-8i+16}$

Simplify.

$\frac{-14+22i}{20}$

Simplify further using the addition of fractions rule, then factor the i out of the 2nd fraction.

$-\frac{7}{10}+\frac{11}{10}i$

Divide.

$\frac{4-i^3}{5+i}$

$\frac{21}{26}+\frac{i}{26}$

$\frac{26}{21}+\frac{26}{i}$

$\frac{21}{26}-\frac{i}{26}$

$\frac{7}{13}-\frac{21i}{6}$

Explanation

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

To find the conjugate, just change the sign in the denominator. The conjugate used will be .

$\left(\frac{4-i^3}{5+i}\right)\left(\frac{5-i}{5-i}\right)$

Now, distribute and simplify.

$\left(\frac{4-i^3}{5+i}\right)\left(\frac{5-i}{5-i}\right)=\frac{20-4i-5i^3+i^4}{25-i^2}$

Recall that $i^2=-1, i^3=-i,\text{ and }i^4=1$

$\frac{20-4i-5i^3+i^4}{25-i^2}=\frac{20-4i-5(-i)+1}{25-(-1)}=\frac{21+i}{26}=\frac{21}{26}+\frac{i}{26}$

Divide.

$\frac{5-2i}{3-8i}$

$\frac{31}{73}+\frac{34i}{73}$

$\frac{31}{73}-\frac{34i}{73}$

$\frac{73}{31}-\frac{73}{64i}$

$\frac{21}{29}+\frac{6i}{29}$

Explanation

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

To find the conjugate, just change the sign in the denominator. The conjugate used will be .

$\left(\frac{5-2i}{3-8i}\right)\left(\frac{3+8i}{3+8i}\right)$

Now, distribute and simplify.

$\left(\frac{5-2i}{3-8i}\right)\left(\frac{3+8i}{3+8i}\right)=\frac{15+34i-16i^2}{9-64i^2}$

Recall that

$\frac{15+34i-16i^2}{9-64i^2}=\frac{15+34i-16(-1)}{9-64(-1)}=\frac{31+34i}{73}=\frac{31}{73}+\frac{34i}{73}$

Divide:

$\frac{7+i}{2-i}$

$\frac{13}{5}+\frac{9i}{5}$

$\frac{5}{13}+\frac{5}{9i}$

$\frac{13}{5}-\frac{9i}{5}$

$\frac{15}{2}+3i$

Explanation

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

To find the conjugate, just change the sign in the denominator. The conjugate used will be .

$\left(\frac{7+i}{2-i}\right)\left(\frac{2+i}{2+i}\right)$

Now, distribute and simplify.

$\left(\frac{7+i}{2-i}\right)\left(\frac{2+i}{2+i}\right)=\frac{14+9i+i^2}{4-i^2}$

Recall that

$\frac{14+9i+i^2}{4-i^2}=\frac{14+9i+(-1)}{4-(-1)}=\frac{13+9i}{5}=\frac{13}{5}+\frac{9i}{5}$

What is $\frac{(i^2)}{1-i}$ ?

$-1+\frac{1}{1-i}$

$1+\frac{i}{i^2}$

Explanation

Since ,

the problem becomes,

$\frac{-1}{1-i}$

$=-1+\frac{1}{1-i}$

Divide.

$\frac{-9+i}{1-i}$

Explanation

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

To find the conjugate, just change the sign in the denominator. The conjugate used will be .

$\left(\frac{-9+i}{1-i}\right)\left(\frac{1+i}{1+i}\right)$

Now, distribute and simplify.

$\left(\frac{-9+i}{1-i}\right)\left(\frac{1+i}{1+i}\right)=\frac{-9-8i+i^2}{1-i^2}$

Recall that

$\frac{-9-8i+i^2}{1-i^2}=\frac{-9-8i+(-1)}{1-(-1)}=\frac{-10-8i}{2}=-5-4i$

Divide.

$\frac{4+3i}{6-i}$

$\frac{21}{37}+\frac{22i}{37}$

$\frac{7}{9}-\frac{22i}{27}$

$\frac{9}{7}+\frac{27}{22i}$

$\frac{2}{3}+\frac{i}{9}$

Explanation

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

To find the conjugate, just change the sign in the denominator. The conjugate used will be .

$\left(\frac{4+3i}{6-i}\right)\left(\frac{6+i}{6+i}\right)$

Now, distribute and simplify.

$\left(\frac{4+3i}{6-i}\right)\left(\frac{6+i}{6+i}\right)=\frac{24+22i+3i^2}{36-i^2}$

Recall that

$\frac{24+22i+3i^2}{36-i^2}=\frac{24+22i+3(-1)}{36-(-1)}=\frac{21+22i}{37}$

Divide.

$\frac{5+i}{3+i}$

$\frac{8}{5}-\frac{i}{5}$

$\frac{8}{5}+\frac{i}{5}$

$\frac{5}{8}+\frac{i}{5}$

$\frac{3}{8}-\frac{i}{4}$

Explanation

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

To find the conjugate, just change the sign in the denominator. The conjugate used will be .

$\left(\frac{5+i}{3+i}\right)\left(\frac{3-i}{3-i}\right)$

Now, distribute and simplify.

$\left(\frac{5+i}{3+i}\right)\left(\frac{3-i}{3-i}\right)=\frac{15-2i-i^2}{9-i^2}$

Recall that

$\frac{15-2i-i^2}{9-i^2}=\frac{15-2i-(-1)}{9-(-1)}=\frac{16-2i}{10}=\frac{8}{5}-\frac{i}{5}$

Divide.

$\frac{5+i}{i}$

Explanation

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

To find the conjugate, just change the sign in the denominator. The conjugate used will be .

$\left(\frac{5+i}{i}\right)\left(\frac{-i}{-i}\right)$

Now, distribute and simplify.

$\left(\frac{5+i}{i}\right)\left(\frac{-i}{-i}\right)=\frac{-5i-i^2}{-i^2}$

Recall that

$\frac{-5i-i^2}{-i^2}=\frac{-5i-(-1)}{-(-1)}=\frac{1-5i}{1}=1-5i$

Divide:

$\frac{4+5i}{5-i}$

$\frac{15}{26}+\frac{29i}{26}$

$\frac{15}{26}-\frac{29i}{26}$

$\frac{29}{26}+\frac{15i}{26}$

$\frac{5}{8}+\frac{29i}{24}$

Explanation

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

To find the conjugate, just change the sign in the denominator. The conjugate used will be .

$\left(\frac{4+5i}{5-i}\right)\left(\frac{5+i}{5+i}\right)$

Now, distribute and simplify.

$\left(\frac{4+5i}{5-i}\right)\left(\frac{5+i}{5+i}\right)=\frac{20+29i+5i^2}{25-i^2}$

Recall that

$\frac{20+29i+5(-1)}{25-(-1)}=\frac{15+29i}{26}=\frac{15}{26}+\frac{29i}{26}$

Find the Quotient of Complex Numbers

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