Question

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

Answer 1

`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)`

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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)`

Was there a typo in the question?