/sci/ - Science & Math

>>10830327

>>10504928

This is an especially stupid question. How do I factor 0.03x^3-4.5x^2+225x+250 without using a calculator or pic related? The rational zeros theorem doesn't seem to help (or I'm using it wrong).

>>10073944

>>9699054
It's just the cubic equation dude.

>>8797420

>>8485513

>>8447374

>>8429725
well there is, pic related

>>8429700
easiest is to use numerical methods to find one root, then factorize for the other 2

>>7970805

Small pleasures thread? Small pleasures thread.

>never actually learned the cubic formula and its general solutions in uni, skirted by the subject but never drilled in, figure I should self-teach it on general principles
>study the problem and its history, come up with an ugly form with a radical in a denominator (my autism won't stand for this), want to get a nicer form
>noted racist and all-around cool dude John Derbyshire is on my shelf, flip through his book (Unknown Quantity), recall just the hint I needed
>persevere, build up a series of relationships between various terms, and eventually fully appreciate the truth of pic related

Or, more succinctly then even wiki's treatment, we may denote all three of the roots of the general cubic equation as follows:

<span class="math"> \displaystyle ax^3 + bx^2 + cx + d = 0 \rightarrow x= -\frac{1}{3a} \bigg( uv + \bar{u}\bar{v} + b \bigg) [/spoiler]

Where v and v-bar correspond to the large radical forms in pic related (differing only by a sign in the appropriate spot), u denotes /any one of the three/ third roots of unity, and u-bar denotes that given root's complex conjugate (which is itself a third root of unity). So I've simply ended up saying the same thing that Cardano did.

I would imagine that this holds good once we entertain complex a,b,c,d, but I'll have to check that out for myself (I don't know it for a fact).

Makes me want to get Ars Magna, and eventually tackle the quartic to complete the autism. One thing at a time however.

>>7363917

>>7124152
Cubic equation

>>6937180
cubic equation

>>6878086