Quadratic Formula and Cubic Formula

This brought back memories of my engineering-school days, where were shown how to calculate the quadratic equation's roots on a slide-rule with the minimum number of 'slides' of the rule.

First write the equation so that the coefficient of x equals 1 (actually dividing both sides by a), i.e.,

x^2 + px + q = 0, where p = b/a and q = c/a.

The two roots are then:

x = -(p/2) ± √[(p/2)^2-q]

The √[…] means square root - sorry, did not have an equation editor handy.

The trick lies in having (p/2) twice and q on its own.

Cool point, I like simple disscusions that could have complicated answers. Another question is which equation suits which data set "best". These may not be the standard forms accepted by the respective science field for publication and are therefore overlooked. I havent bothered with the science of arithmetic since I discovered TK Solver, MathCad and , CurvExpert. I quess it is because that far beyond being curious about arithmetic, I jealously gaurd my time so that family and football can be important to me. Arithmetic for the sake of arithmetic is as consuming and productive as playing games on the internet. This is a handy tool for pump calibration and is best solved using TK Solver.

GPM=

(A*(RPM/RD))-((B*L^C)*((RD/RPM)^((2*C)-1)))

RD= Design RPM

RPM= Actual Pump RPM L= Static Head ("Lift")

You wrote:

"Arithmetic for the sake of arithmetic is as consuming and productive as playing games on the internet."

I'm not sure what you mean by Arithmetic for the sake of arithmetic, but I'm with you on Football. The reason I learn derivations is so I know what I"m talking about. It's been my experience that it takes the real world about 5 minutes present a problem your software can't handle. If you understand where the formula came from in the first place, or the math behind a computer program, you can more often than not figure out a work around.

Its algbra of first grade in middle school. you hve several solutions method except above mehod.

if you want to solve it with slide rule. it will use Weda therom. the roots can be read easily on the rule scale.

My slide rule still sleeps in somewhere of my room.

Hi Roger

As a curiosity, here is a variant of the cubic equation solution you gave. I've used this algorithm to solve a cubic equation for the turning points of an elliptical orbit around a black hole.

This is a box in a chapter named "Orbital equations" on my website and eBook Relativity 4 Engineers.

Cubic Equations.

I was taught a solution using trig identities which worked fine for my purpose (and had the bonus that I could follow its derivation ).However it fell down (at least for me !) with (-1)^0.5. I later found another method which worked well in all case (I don't think it was a re-arrangement of the method you quote) , though no derivation was given. Are ther any neat tricks to get around the 'nasty' roots with i ?

I found it far more useful to get comfortable with complex numbers. They're so useful in so many ways (such as reducing long and cumbersome trig problems to simple algebra) that I don't know how I got on without them.

The 'i' in the root(s) is simply telling you that the solution is a perfectly ordinary complex number, ie, the root does not lie on the Real axis but is above or below it.