SOLUTION: Hi there, I'm confused on the following problem and could use some help please: I need to multiply and simplify (5 + 7i)^2. Am I going to multiply 5 x 5 then 7i x 7i? I woul

Algebra ->  Expressions-with-variables -> SOLUTION: Hi there, I'm confused on the following problem and could use some help please: I need to multiply and simplify (5 + 7i)^2. Am I going to multiply 5 x 5 then 7i x 7i? I woul      Log On


   

Question 1040428: Hi there, I'm confused on the following problem and could use some help please:
I need to multiply and simplify (5 + 7i)^2. Am I going to multiply 5 x 5 then 7i x 7i?
I would really appreciate some help, thank you!
Found 2 solutions by jim_thompson5910, ikleyn:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
You're on the right track. You're missing 2 pieces though

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(5 + 7i)^2 is the same as (5 + 7i)*(5 + 7i)

It's like saying x^2 is the same as x*x

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Let's use FOIL to expand out (5 + 7i)*(5 + 7i)

First terms: 5*5 = 25
Inner terms: 7i*5 = 35i
Outer terms: 5*7i = 35i
Last terms: 7i*7i = 49i^2

So we now add up all the terms to get

25+35i+35i+49i^2

25+70i+49i^2 ... combine like terms

25+70i+49(-1) ... replace i^2 with -1. See note

25+70i-49

-24+70i ... combine like terms


Note: since we have defined i = sqrt(-1), squaring both sides leads to i^2 = -1

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The final answer is -24+70i

Answer by ikleyn(52817) About Me  (Show Source):
You can put this solution on YOUR website!
.
I need to multiply and simplify (5 + 7i)^2. Am I going to multiply 5 x 5 then 7i x 7i?
I would really appreciate some help, thank you!
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

1.  What you are planning to do is incorrect.


2.  Do you know this formula: %28a+%2B+b%29%5E2 = a%5E2+%2B+2ab+%2B+b%5E2 ?

    It is applicable to complex numbers exactly as it is applicable to real numbers.
    (It is a general formula !)

3.  So, you need to take 5%5E2, 2%2A5%2A7i and %287i%29%5E2 and add up these addends. You will get

    25+%2B+70%2Ai+%2B+49i%5E2.

    Now, you should remember that i%5E2 = -1. So your sum is

    25 + 70i -49 = -24 + 70i.


It is your answer.

Answer.  %285+%2B+7i%29%5E2 = -24 + 70i.

On complex numbers see the lessons
    - Complex numbers and arithmetic operations on them
    - Complex plane
    - Addition and subtraction of complex numbers in complex plane
    - Multiplication and division of complex numbers in complex plane
    - Raising a complex number to an integer power
    - How to take a root of a complex number
    - Solution of the quadratic equation with real coefficients on complex domain
    - How to take a square root of a complex number
    - Solution of the quadratic equation with complex coefficients on complex domain
in this site.

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