How to factorize the quadratic $a(b-c)x^2 + b(c-a)x + c(b-a) = 0$?

Question

How do i factorize this equation: $a(b-c)x^2 + b(c-a)x + c(b-a) = 0$

I tried the quadratic formula, but the discriminant is not factorising into a perfect square.
Please help!

Answer 1

HINT:

Observe that $$a(b-c)-c(b-a)=-b(c-a)$$

Put the value of $b(c-a),$ and take out common from the first two terms & the last two terms and see what happens?

Answer 2

When $a=1,b=2,c=3$ you obtain $-x^2+4x+3$, whose discriminant is $28$, a nonsquare. When $a=1,b=2,c=1$ you obtain $x^2+1$, whose discriminant is $-1$.

Conclusion : the discriminant is not a perfect square in general, it may be positive or negative. There is no factorisation (unless there is a typo in the OP).