Why are complex roots of quadratic equations conjugate pairs?

[thereddevils]

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]

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

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?

 

[thereddevils]

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?

thanks Mark!