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For the implicit equation,

${\left({\mathrm{ax}}^{3}+{\mathrm{bx}}^{2}+{\mathrm{cx}}^{1}+d\right)}^{3}={\left({b}^{2}-\mathrm{4ac}\right)}^{1/2}$,

numerical methods will be needed.

Below is an equation with a radical relating to pentagonal symmetry:

$\sqrt{3}\sqrt{5-2\sqrt{5}}$ inscribed sphere = 1/circumscribed sphere $\frac{\sqrt{5+2\sqrt{5}}}{\sqrt{\mathrm{15}}}$

${\left({\mathrm{ax}}^{3}+{\mathrm{bx}}^{2}/\left({\mathrm{ax}}^{3}+{\mathrm{bx}}^{2}\right)\right)}^{3}={\left({b}^{2}-\mathrm{4ac}\right)}^{1/2}$

$\frac{2b}{\sqrt{5}}+\sqrt{5}+1$

$\frac{a\frac{e^{-y}-1}{e^{\mathrm{-y}}}}{\frac{c+dy}{1+c/b-d\frac{e^{\mathrm{-y}}-1}{gb}}+\frac{1}{e^{\mathrm{-y}}}}$

If your browser displays those equations correctly, it is enabled to display Design Science, Presentation, MathPlayer, ... MathML.

the original

METHODS TO WRITE PORTABLE and PRESENTABLE EQUATIONS WITHOUT BITMAPS or PROPRIETARY TOOLS

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