The discussion centers on demonstrating that if one root of the quadratic equation \( ax^2 + bx + c = 0 \) is \( m + ni \), then the other root must be \( m - ni \). This is established through the quadratic formula, where the roots can be expressed as \( (x - (m + ni))(x - (m - ni)) = 0 \). The participants emphasize the need to expand this expression correctly to confirm that both roots satisfy the equation, ultimately leading to the conclusion that \( m - ni \) is indeed a valid root.
PREREQUISITESStudents studying algebra, mathematics educators, and anyone interested in understanding the properties of complex roots in quadratic equations.
The last equation above is not correct. For one thing, there should be an x2 term.thereddevils said:Homework Statement
If one of the roots of the quadratic equation, ax^2+bx+c=0 is m+ni, show that the other root is m-ni.
Homework Equations
The Attempt at a Solution
How do i actually show this? I mean it's a well known fact and a direct outcome of the quadratic formula. Is this valid?
(x-(m+ni))(x-(m-ni))=0
then x-2m+m^2+n^2=0
Mark44 said:The last equation above is not correct. For one thing, there should be an x2 term.
You are given that m + ni is a root of the equation ax2 + bx + c = 0, so it should be true that a(m + ni)2 + b(m + ni) + c = 0.
Expand the stuff on the left and you will get a complex number that must be zero.
Now, what do you need to do to show that m - ni is also a solution of the same quadratic equation?