Question 553408: The Question is:
Given a Quadratic equation with rational coefficients with a root of 6+5i
a) What is another root to this equation?
b) What are the factors of the equation?
c) What is the quadratic equation with the given root?
d) What is the representative function?
I know this is long but i tried and my teacher took up my references to grade them, and I don't remember any of the rules. I think answer a is 6-5i. Can you please help? This is a very important grade and I have no room for mistakes. Thank You for any help :)
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! a) What is another root to this equation?
If a+bi is one root, then the conjugate a-bi is another root
So 6 - 5i is another root
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b) What are the factors of the equation?
If p and q are roots, then x-p and x-q are factors
So if 6+5i and 6-5i are roots, then x-(6+5i) and x-(6-5i) are factors
They can be simplified to x-6-5i and x-6+5i
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c) What is the quadratic equation with the given root?
Multiply the factors to get
(x-6-5i)(x-6+5i)
((x-6)-5i)((x-6)+5i)
(x-6)^2-(5i)^2 ... Note: I'm using the difference of squares formula here
(x-6)^2-25i^2
(x-6)^2-25(-1)
(x-6)^2+25
x^2-12x+36+25
x^2-12x+61
So the quadratic equation with the given root is x^2-12x+61 = 0
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d) What is the representative function?
The function is f(x) = x^2-12x+61
All we're doing here is replacing the =0 part with f(x) = ...
If you need more help, email me at jim_thompson5910@hotmail.com
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Jim
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