SOLUTION: Find the quadratic equation with roots -1+4i and -1-4i. The correct answer is x^2 + 2x + 17 = 0 according to my teacher. Is this answer wrong because everytime I try to solve the

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Question 120039: Find the quadratic equation with roots -1+4i and -1-4i.
The correct answer is x^2 + 2x + 17 = 0 according to my teacher.
Is this answer wrong because everytime I try to solve the problem, I do not get that answer.
If it is right, please explain how.
Answer by bucky(2189) (Show Source):
You can put this solution on YOUR website!
You are given that the roots of a quadratic equation are:
.
and
.
and you are asked to find the quadratic equation that has those roots.
.
Since these are roots, you know that they are the values of x that solve a quadratic
expression that is equal to zero. Therefore, you know that:
.

.
is one answer and
.

.
is the other answer.
.
You can convert these two answers to the factors of the quadratic by getting everything
on the right side of these two equalities. So let's start with:
.

.
and subtract from both sides. When you do that subtraction, you get:
.

.
This tells you that (x + 1 - 4i) is one of the factors of the quadratic.
.
Similarly, next go to the other root:
.

.
Convert it to a factor by subtracting from both sides. When you do that subtraction
you get:
.

.
and this tells you that (x + 1 + 4i) is the other factor of the quadratic.
.
To get the quadratic equation, multiply these two factors together and set the product equal
to zero.
.
So you multiply (x + 1 - 4i) times (x + 1 + 4i) and set the result equal to zero.
.
This multiplication requires that you multiply each term in the second set of parentheses
by each term in the first set of parentheses and then combine terms. So let's take the x
from the first set of parentheses and multiply it times everything in the second set of parentheses.
In other words multiply:
.
to get
.
Next take the +1 from the first set of parentheses and multiply it times everything in
the second set of parentheses:
.
to get
.
Finally take the -4i from the first set of parentheses and multiply it times everything in
the second set of parentheses:
.
to get
.
In the third term of this last product, note that by definition and if in
you replace by -1 you get . So the third multiplication
for which you got can be simplified to
.
Now combine all three of the products by adding them:
.

.
Group the common terms and you get:
.

.
And when you combine the grouped terms some of them cancel out. You are left with:
.

.
In the last step you set this result equal to zero and you have as the quadratic equation:
.

.
Hope this helps you to find your error. In this case your teacher is correct.
.