Find the exact solution of cubic equation 0 = x^3 - 1.5*x^2 + 0.5*x - 0.75/(12*SQRT(3))

[pascalswager]

1. The cubic equation 0 = x^3 - 1.5*x^2 + 0.5*x - 0.75/(12*SQRT(3)) has three real roots, which can easily be found by iteration:
x≈ 0.100368...
x≈ 0.338955...
x≈ 1.06068...

Can you find the EXACT representation of the first root? (x≈0.100368...) ?

2. More generally, can you find an exact representation of the lowest root for the following equation?
0 = x^3 - 1.5*x^2 + 0.5*x - t/(12*SQRT(3)) for 0<t<1?

Any help much appreciated!

 

[Deleted member 4993]

1. The cubic equation 0 = x^3 - 1.5*x^2 + 0.5*x - 0.75/(12*SQRT(3)) has three real roots, which can easily be found by iteration:
x≈ 0.100368...
x≈ 0.338955...
x≈ 1.06068...

Can you find the EXACT representation of the first root? (x≈0.100368...) ?

2. More generally, can you find an exact representation of the lowest root for the following equation?
0 = x^3 - 1.5*x^2 + 0.5*x - t/(12*SQRT(3)) for 0<t<1?

Any help much appreciated!

It is tedious - but:

For a cubic equation ax^3+bx^2+cx+d=0ax3+bx2+cx+d=0, let p,q, and r be its roots, then the following holds:

Root expression	Equals to
p + q + r	-b/a
pq + qr + rp	c/a
pqr	-d/a

ref: https://brilliant.org/wiki/cubic-equations/

You could also use Cardano's method - ref: https://brilliant.org/wiki/cardano-method/

Please let us know if you are still stuck.