Are there any Math fans here ?

[Heron]

I'm not a numbers person, I'm a word person.
Which is why MATH as opposed to MATHS drive's me nuts!
How can mathematics (plural) become shortened to math?
American English is in itself an oxymoron!
Sorry..... Just had to get that off my chest!
No offence meant to all you Americans

As a neutral Swede I see mathematics shortened to maths which is then properly shortened to math; all meaning the same. (whatever mathematics may be to people, I doubt any Vulcan besides me wouldn't agree)
For example I guess you write 10 m (ten metres) instead of 10 ms, so I think it is OK to cut off the trailing "s" for those who want to do that.
But this is of course just my point of view – me usually being wrong.

 

[Heron]

No one doubts solutions of easy equations, but what about those of x⁵+2x⁴+3x³+4x²+5x+3 = 0
? Talking about real solutions.

 

[vangelis]

x = -1

minus is real in my world!

 

[Heron]

x = -1

minus is real in my world!

Perfectly right! The question then, are there any other real solutions of the equation? (for people not comfortable with -1)

Answer: No, there is no other solution than Vangelis'. (+)

By the way, about quartic equations
–2x⁴ + 5x³ – 9x² + 8x + 5 = 0

Well, if we let a = –2, b = 5, c = –9, d = 8, and e= 5; then the fascinating quartic solution formula gives something like
x[1]=((((((8*a*c-3*b^2)/(8*a^2))*((-3*b^4+16*a*(a*(16*a*e-4*b*d)+b^2*c))/(256*a^4)-((8*a*c-3*b^2)/(8*a^2))^2/36)/(-3)+((b^2*b+4*a*(2*a*d-b*c))/(8*a^3))^2/8)/2-((((8*a*c-3*b^2)/(8*a^2))*((-3*b^4+16*a*(a*(16*a*e-4*b*d)+b^2*c))/(256*a^4)-((8*a*c-3*b^2)/(8*a^2))^2/36)/3-((b^2*b+4*a*(2*a*d-b*c))/(8*a^3))^2/8)^2/4-(((8*a*c-3*b^2)/(8*a^2))^2/12+(16*a*(a*(16*a*e-4*b*d)+b^2*c)-3*b^4)/(256*a^4))^3/27)^0,5)^(1/3)+((((8*a*c-3*b^2)/(8*a^2))*((-3*b^4+16*a*(a*(16*a*e-4*b*d)+b^2*c))/(256*a^4)-((8*a*c-3*b^2)/(8*a^2))^2/36)/(-3)+((b^3+4*a*(2*a*d-b*c))/(8*a^3))^2/8)/2+((((8*a*c-3*b^2)/(8*a^2))*((-3*b^4+16*a*(a*(16*a*e-4*b*d)+b^2*c))/(256*a^4)-((8*a*c-3*b^2)/(8*a^2))^2/36)/3-((b^3+4*a*(2*a*d-b*c))/(8*a^3))^2/8)^2/4+((3*b^4-16*a*(a*(16*a*e-4*b*d)+b^2*c))/(256*a^4)-((8*a*c-3*b^2)/(8*a^2))^2/12)^3/27)^0,5)^(1/3)-((8*a*c-3*b^2)/(8*a^2))/3)/2)^0,5-0,5*(-2*(((((8*a*c-3*b^2)/(8*a^2))*((-3*b^4+16*a*(a*(16*a*e-4*b*d)+b^2*c))/(256*a^4)-((8*a*c-3*b^2)/(8*a^2))^2/36)/(-3)+((b^3+4*a*(2*a*d-b*c))/(8*a^3))^2/8)/2-((((8*a*c-3*b^2)/(8*a^2))*((-3*b^4+16*a*(a*(16*a*e-4*b*d)+b^2*c))/(256*a^4)-((8*a*c-3*b^2)/(8*a^2))^2/36)/3-((b^3+4*a*(2*a*d-b*c))/(8*a^3))^2/8)^2/4-((((8*a*c-3*b^2)/(8*a^2))^2/12+(-3*b^4+16*a*(a*(16*a*e-4*b*d)+b^2*c))/(256*a^4)))^3/27)^0,5)^(1/3)+(((8*a*c-3*b^2)/(8*a^2)*((-3*b^4+16*a*(a*(16*a*e-4*b*d)+b^2*c))/(256*a^4)-((8*a*c-3*b^2)/(8*a^2))^2/36)/3-((b^3+4*a*(2*a*d-b*c))/(8*a^3))^2/8)/(-2)+((((8*a*c-3*b^2)/(8*a^2))*((-3*b^4+16*a*(a*(16*a*e-4*b*d)+b^2*c))/(256*a^4)-((8*a*c-3*b^2)/(8*a^2))^2/36)/3-((b^3+4*a*(2*a*d-b*c))/(8*a^3))^2/8)^2/4-((((8*a*c-3*b^2)/(8*a^2))^2/12+(-3*b^4+16*a*(a*(16*a*e-4*b*d)+b^2*c))/(256*a^4)))^3/27)^0,5)^(1/3)-((8*a*c-3*b^2)/(8*a^2))/3)-2*((8*a*c-3*b^2)/(8*a^2))-((b^2*b+4*a*(2*a*d-b*c))/(8*a^3))/((((((8*a*c-3*b^2)/(8*a^2))*((-3*b^4+16*a*(a*(16*a*e-4*b*d)+b^2*c))/(256*a^4)-((8*a*c-3*b^2)/(8*a*a))^2/36)/3-((b^3+4*a*(2*a*d-b*c))/(8*a^3))^2/8)/(-2)-((((8*a*c-3*b^2)/(8*a^2))*((-3*b^4+16*a*(a*(16*a*e-4*b*d)+b^2*c))/(256*a^4)-((8*a*c-3*b^2)/(8*a^2))^2/36)/3-((b^3+4*a*(2*a*d-b*c))/(8*a^3))^2/8)^2/4-((((8*a*c-3*b^2)/(8*a^2))^2/12+(-3*b^4+16*a*(a*(16*a*e-4*b*d)+b^2*c))/(256*a^4)))^3/27)^0,5)^(1/3)+((((8*a*c-3*b^2)/(8*a^2))*((-3*b^4+16*a*(a*(16*a*e-4*b*d)+b^2*c))/(256*a^4)-((8*a*c-3*b^2)/(8*a^2))^2/36)/3-((b^3+4*a*(2*a*d-b*c))/(8*a^3))^2/8)/(-2)+((((8*a*c-3*b^2)/(8*a^2))*((-3*b^4+16*a*(a*(16*a*e-4*b*d)+b^2*c))/(256*a^4)-((8*a*c-3*b^2)/(8*a^2))^2/36)/3-((b^3+4*a*(2*a*d-b*c))/(8*a^3))^2/8)^2/4-((((8*a*c-3*b^2)/(8*a^2))^2/12+(-3*b^4+16*a*(a*(16*a*e-4*b*d)+b^2*c))/(256*a^4)))^3/27)^0,5)^(1/3)-(8*a*c-3*b^2)/(8*a^2)/3)/2)^0,5)^0,5-b/(4*a)=-0.399381267914565…

Accordingly x[2] = 1.72373826664918…
x[3] and x[4] are not real but complex solutions (0,587822 ± 1.812712i).
As simple as that

 

[Heron]

I know this was probably backwoods stuff, but neither the less it may be of importance also for others …
– just nod if you hear me –

 

[Aeolienne]

I know this was probably backwoods stuff, but neither the less it may be of importance also for others …
– just nod if you hear me –

What's with the Vangelis reference?

 

[Heron]

What's with the Vangelis reference?

Yes, Vangelis is great
Also, I find it astonishing that
(9√6 + 19)^⅓ – (9√6 – 19)^⅓ = 2
I wouldn't expect the difference on the left hand side to be an integer but it is.
An experienced solver of cubic equations may recognize it as the solution of the cubic' x³ + 15x – 38 = 0 (with solution x = 2), but still it feels strange.
_____
Now I think I see my error: (9√6 + 19)^⅓ and (9√6 – 19)^⅓ are one by one both algebraic numbers far from the world of integers but together they may compose a product (9√6 + 19)^⅓ · (9√6 – 19)^⅓ = 5. Therefore, together they are clearly related to the world of integers.