SOLUTION: Could someone help me with these, I had it last year but forgot how it works: 1) Find a quadratic equation with roots (4+i)and(4-i). 2) Given the equation x^3-2x^2+x-3=0, what

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Question 198572: Could someone help me with these, I had it last year but forgot how it works:
1) Find a quadratic equation with roots (4+i)and(4-i).
2) Given the equation x^3-2x^2+x-3=0, what are the possible rational roots??
3) Find the irratinal roots of the equation; x^3-4x^2+2x+1=0 I should use the quadratic formula to solve the equation, it might sound stupid but I never really understood how to use the quadratic formula =(

Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!
or Start with the given solutions.

or Subtract 4 from both sides (for each equation).

or Square both sides

or Square i to get -1 and square -i to get -1

or Add 1 to both sides.

Since the equations are the same, we can focus on one equation:

FOIL

Combine like terms.

So the quadratic with the roots 4+i and 4-i is

# 2

Any rational zero can be found through this equation

where p and q are the factors of the last and first coefficients

So let's list the factors of -3 (the last coefficient):

Now let's list the factors of 1 (the first coefficient):

Now let's divide each factor of the last coefficient by each factor of the first coefficient

Now simplify

These are all the distinct rational zeros of the function that could occur

# 3

First, let's find the possible rational zeros

Any rational zero can be found through this equation

where p and q are the factors of the last and first coefficients

So let's list the factors of 1 (the last coefficient):

Now let's list the factors of 1 (the first coefficient):

Now let's divide each factor of the last coefficient by each factor of the first coefficient

Now simplify

These are all the distinct rational zeros of the function that could occur

Let's see if the possible zero is really a root for the function

So let's make the synthetic division table for the function given the possible zero :

1 | 1 -4 2 1
| 1 -3 -1
1 -3 -1 0

Since the remainder (the right most entry in the last row) is equal to zero, this means that is a zero of

Take note that the first three values in the bottom row are 1, -3, and -1. So this means that

Now all we need to do is solve to find the next two zeros:

Notice we have a quadratic in the form of where , , and

Let's use the quadratic formula to solve for "x":

Start with the quadratic formula

Plug in , , and

Negate to get .

Square to get .

Multiply to get

Rewrite as

Add to to get

Multiply and to get .

or Break up the expression.

So the next two zeros are or

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Answer:

So the three roots are , or

where the irrational roots are and