how to factor m^2+mn+n^2

Question

is the equation m^2+mn+n^2 factorable?

Answer 1

This doesn't factorise into real factors, but this is how you could attempt to factorise it as a quadratic.

Let m²+mn+n²=0, then:

m²+mn=-n²,

m²+mn+¼n²=¼n²-n²,

(m+½n)²=-¾n².

We could take the square root of each side, but the RHS is negative and the square root of a negative number is an imaginary number. Imaginary in mathematics means we use i to represent √-1.

So, m+½n=±½ni√3, making m=-½n±½ni√3.

Now we can factorise: (m+½n+½ni√3)(m+½n-½ni√3). These are called complex factors.

m²+mn+n²=(m³-n³)/(m-n).