Mathematician Finds Easier Way to Solve Quadratic Equations

[Gerson W. Barbosa]

(05-05-2024 02:24 AM)Namir Wrote:  Let's keep in mind the KISS principle (keep it simple stupid), or what Einstein said, "Make it simple, but no simpler". Using fancy functions like cosh, exp, sin, and so on is fancy, but introduces a bit of complicatrion in my very humble opinion.

I fear you have completely missed the point. We could, for instance, apply the identity \(\arcsin{z}=\arccos{\sqrt{1-z^2}}\) to

\[x_{1}=\left({\cos\left({\arcsin{\frac{\sqrt{q}}{-\frac{p}{2}}}}\right)+1}\right)\times-\frac{p}{2}\]
and easily take the trigonometric functions out of the formula. But then we would get back to the original pq formula, which as we know cannot handle, on a 12-digit calculator, the first example in Palmer’s article, a=1E-13, b=-2 and c=1, whose roots at that precision are 2E13 and 0.5, not 2E13 and zero. Of course that is not a concern on Free42 if we are interested solely in 12-digit answers.

Sure these have some shortcomings, like b or c having to be different from zero, depending on the formula we choose, or real results being shown with null imaginary parts in certain cases, for example when a=1, b=-1 and c=-2.

The use of transcendental functions are just trade-offs for a little more accuracy while keeping compactness. In a failed attempt (complex number is an invalid type for the LN1+X function), in order to keep the stack register T intact I would use – horror of horrors! – four transcendental functions in a row.

Regards,

Gerson.


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Code:


00 { 26-Byte Prgm }
01▸LBL "Z"
02 X<> ST Z
03 STO÷ ST Z
04 STO+ ST X
05 +/-
06 ÷
07 RCL ST Y
08 SQRT
09 RCL÷ ST Y
10 ASIN
11 COS
12 LN1+X
13 E↑X
14 ×
15 STO÷ ST Y
16 END