SOLUTION: 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?

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
-------------------------------------------------------

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

-------------------------------------------------------

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

-------------------------------------------------------

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

Also, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Thank you

Jim
RELATED QUESTIONS
Given a quadratic equation with rational coefficients with a root of negative 2/3i: (answered by jsmallt9)

Given a quadratic equation with rational coefficients with a root of 5-i: (answered by nerdybill,jsmallt9)

2-5i is one root of a polynomial equation. What is another root that goes along with this (answered by Alan3354)

Write a Quadratic equation with rational coefficients if one root is 4+sqrt( 5 ).... (answered by ReadingBoosters)

Find the polynomial equation with rational coefficients whose root is i-square root of (answered by josgarithmetic)

What is the degree of the simplest polynomial with integger coefficients that has square... (answered by josmiceli)

how do you write a quadratic equation with a root of 2-5i... (answered by robertb)

Write a polynomial function with rational coefficients in standard form with the given... (answered by Fombitz)

Write a polynomial function with rational coefficients in standard form with the given... (answered by MathLover1,KMST)