Using Quadratic formula in other fields

[ramsey2879]

Since the the discovery of the complex number field resulted in part from efforts to resolve the quadratic formula when b^2 - 4ac was negative, am I right to use the quadratic formula to solve zM^2 + wM - t = 0 in the complex number field even when z, w or t can be fractional powers?

 

[HallsofIvy]

"Fractional powers" of what? As long as you have chosen a specific branch so the roots are well defined, they are just numbers. Yes, the quadratic formula applies.

 

[dodo]

Hyper-stupid question: in complex numbers, does the square root have also two values, one plus and one minus? If so, how would the square root of (4 i^2) look like, for example? I ask, of course, because this is present in the usual formula for solving quadratic equations; and all the same if you 'complete the square'.

 

[HallsofIvy]

Yes, in the complex numbers, every number has two square roots, one the additive inverse of the other. I am reluctant to say "one plus and the other minus" since that could be construed to mean "positive and negative" which has no meaning for general complex numbers. There is no such thing as a "positive" complex number unless it happens to be real.

The two square roots of 4i² = -4 are 2i and -2i, of course. A little more complicated question would be the two square roots of 4i. They are \sqrt{2}+ i\sqrt{2} and -\sqrt{2}-i\sqrt{2}.