Warmup

Algebra Level 2

If 25 3 − 4 i = a + b i \dfrac{25}{3-4i}=a+bi , where a a and b b are real numbers and i i is the imaginary unit , find a + b a+b .


The answer is 7.

Akeel Howell by Akeel Howell , 22, USA

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1 solution

Akeel Howell
Feb 9, 2017

25 3 − 4 i = a + b i ⟹ 25 ( 3 + 4 i ) ( 3 − 4 i ) ( 3 + 4 i ) = a + b i So a + b i = 25 ( 3 + 4 i ) 9 − ( − 16 ) ⟹ a + b i = 25 ( 3 + 4 i ) 25 So a + b i = 3 + 4 i ∴ a + b = 3 + 4 = 7 \dfrac{25}{3-4i}=a+bi \implies \dfrac{25(3+4i)}{(3-4i)(3+4i)}=a+bi \\ \text{So} \space \space a+bi = \dfrac{25(3+4i)}{9-(-16)} \implies a+bi = \dfrac{25(3+4i)}{25} \\ \text{So} \space a+bi = 3+4i \\ \therefore a+b = 3+4 = \boxed{7}

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