SOLUTION: 4 - i is a solution of a quadratic equation with real coefficients.Find the other solution. How is this one done?

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Question 1207475: 4 - i is a solution of a quadratic equation with real coefficients.Find the other solution.

How is this one done?
Found 3 solutions by josgarithmetic, ikleyn, math_tutor2020:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
See the other partly solved example.

Answer by ikleyn(52786) About Me  (Show Source):
You can put this solution on YOUR website!
.

The general theorem of algebra states that if a polynomial with real coefficients
has a complex number root  a+bi,  b=/= 0,  then it has another complex number root  a-bi,  too.

According to this theorem,  if  4-i  is the root of your quadratic equation with real coefficients,
then it has the root  4+i,  too.   It is your other root.



Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Answer: 4+i

Quick explanation: If a+bi is one root, then its paired counterpart (known as the complex conjugate) would be a-bi
This applies only when all of the coefficients are real numbers.

Another example: The root 2+i pairs up with 2-i

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Let's say you didn't know about the complex conjugate, or that you might be curious about an alternative pathway.


We can isolate the "i" term and square both sides to generate a quadratic from it.
x+=+4-i

x-4+=+-i

%28x-4%29%5E2+=+%28-i%29%5E2

%28x-4%29%5E2+=+-1

x%5E2-8x%2B16+=+-1

x%5E2-8x%2B16%2B1+=+0

x%5E2-8x%2B17+=+0

Now let's apply the quadratic formula
x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29

x+=+%28-%28-8%29%2B-sqrt%28%28-8%29%5E2-4%281%29%2817%29%29%29%2F%282%281%29%29

x+=+%288%2B-sqrt%2864+-+68%29%29%2F%282%29

x+=+%288%2B-sqrt%28-4%29%29%2F%282%29

x+=+%288%2B-+2i%29%2F%282%29

x+=+%288%2B+2i%29%2F%282%29 or x+=+%288-2i%29%2F%282%29

x+=+4%2Bi or x+=+4-i
We arrive at roots 4+i and 4-i to help confirm the answer.

This is a somewhat long-winded pathway to basically rephrase what I mentioned at the top.
If a+bi is one root, then a-bi is also included in the mix.
This only applies when all coefficients are real numbers.