How could I rewrite $\dfrac{6 + 4 {i}} { -9 - 4 {i}}$ in a+bi form?

Peace to all. When I solve the problem I get $$\dfrac{70} { 97}$$ - $$\dfrac{12{i}} {97}$$ and it's the wrong answer. How exactly do you go about solving this problem?

This is my work: I received that answer by multiplying both the numerator and denominator by the conjugate partner of "-9 - 4i" which is "-9 + 4i".

$$\dfrac{-54 + 24 {i} - 36{i} +16i^2} { 81 - 36 {i} + 36 {i} - 16i^2}$$

Combining like terms: $$\dfrac{-54 + 24 {i} - 36{i} +16(-1)} { 81 - 36 {i} + 36 {i} - 16(-1)}$$ = $$\dfrac{-70 - 12i} { 81 + 16}$$ = $$\dfrac{-70} { 97}$$ - $$\dfrac{12{i}} {97}$$

• Please show all your work. How did you get that number? Oct 5 '21 at 1:32
• You got a sign wrong somewhere. I'm getting that the numerator should be $-70-12i$. Your edited question has the correct answer, which is different from your original answer. $(-9-4i)(-9+4i)=81+16=97$, not $-97$. Oct 5 '21 at 1:43
• @RobertShore As I was writing it I noticed that I didn't copy the correct sign from my work. It's those tiny minute (important) details that I overlook sometimes with math. Thank you Oct 5 '21 at 1:52
• "$\dfrac{70 - 12 {i}} {97}$ or $\dfrac{70{i}} { 97}$ - $\dfrac{12{i}} {97}$". Why did you write "or" in between these expressions? These both expressions are different. $\dfrac{70 - 12 {i}} {97} = \dfrac{70}{97} - \dfrac{12i}{97}$. You wrongly splitted the expression. Additionally I think your intention was to write $\dfrac{-70 -12i}{97}$ not $\dfrac{70-12i}{97}$. Please edit it.
– user947346
Oct 5 '21 at 3:01
• @ProThala Thank you, changes made Oct 5 '21 at 17:52

To avoid confusion, we better take out the negative sign of the denominator first and then multiply the conjugate of 9+4i.

$$\displaystyle {\quad \frac{6+4i}{-9-4i}\\= -\frac{6+4i}{9+4i}\cdot\frac{9-4i}{9-4i} \\=-\frac{70+12i}{81+16} \\=-\frac{70}{97} -\frac{12}{97} i}$$

$$\textrm{ :|D Wish it helps!}$$

• (+1) nice explanation :)
– user947346
Oct 5 '21 at 3:05
• Thank you very much for your appreciation!
– Lai
Oct 5 '21 at 16:17

By multiplying both the numerator and denominator by the conjugate partner of "-9 - 4i" which is "-9 + 4i", you will get:

$$\dfrac{-54 + 24 {i} - 36{i} +16i^2} { 81 - 36 {i} + 36 {i} - 16i^2}$$

Then combining like terms:

$$\dfrac{-54 + 24 {i} - 36{i} +16(-1)} { 81 - 36 {i} + 36 {i} - 16(-1)}$$ = $$\dfrac{-70 - 12i} { 81 + 16}$$ = $$\dfrac{-70} { 97}$$ - $$\dfrac{12{i}} {97}$$

Hope this assists anyone who has/had similar difficulties in solving an equation similar to or like the one above.

• @Moo Thank you, changes made Oct 5 '21 at 17:51