How do you factor #5a^2 - ab + 6b^2#?

1 Answer
George C.
Jul 5, 2016

#5a^2-ab+6b^2=1/20(10a-(1+sqrt(119)i)b)(10a-(1-sqrt(119)i)b)#

Explanation:

#5a^2-ab+6b^2#

The discriminant #Delta# is negative:

#Delta = (-1)^2-4(5)(6) = 1-120 = -119#

So this quadratic can only be factored using Complex coefficients:

#5a^2-ab+6b^2#

#=1/20(100a^2-20ab+120b^2)#

#=1/20((10a-b)^2+119b^2)#

#=1/20((10a-b)^2-(sqrt(119)i b)^2)#

#=1/20((10a-b)-sqrt(119)i b)((10a-b)+sqrt(119)i b)#

#=1/20(10a-(1+sqrt(119)i)b)(10a-(1-sqrt(119)i)b)#

#color(white)()#
Footnote

If the last sign in the quadratic was a minus then this would be much simpler:

#5a^2-ab-6b^2=(5a-6b)(a+b)#

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