Question

Va rog sa rezolvati ecuatia de gradul 3:


$\displaystyle\\4x^3 - 3x -\frac{1}{2} =0$


Are 3 solutii reale din care 2 solutii negative si una pozitiva.

Ma intereseaza in special solutia pozitiva.

Mi-ar prinde bine si daca imi dati o idee.

Answer 1

4x³-3x-1/2=0

a=4;b=0;c=-3;d=-1/2

p=c/a-b²/3a²=-3/4-0²/3*4²=-3/4-0=-3/4

q=2b³/27a³-bc/3a³+d/a=2*0³/27*4³-0*(-3)/3*4³+(-1/2)/4=0-0-1/8=-1/8

Δ=(p/3)³+(p/2)²=(-3/4/3)³+(-3/4/2)²=-1/64+9/64=8/64=1/8

P=∛{-q/2+√[(p/3)³+(p/2)²]}=∛[1/16+√(1/8)]=∛(0,0625+0,35)=∛0,4125=0,744

Q=∛{-q/2-√[(p/3)³+(p/2)²]}=∛[1/16-√(1/8)]=∛(0,0625-0,35)=-∛0,2875=-0,66

y₁=P+Q=0,744-0,66=0,084

y₂=-(P+Q)/2+i√3(P-Q)/2=-(0,744-0,66)/2+i√3(0,744+0,66)/2=-0,042+1,404i√3/2=-0,042+0,702i√3

y₃=-(P+Q)/2-i√3(P-Q)/2=-(0,744-0,66)/2-i√3(0,744+0,66)/2=-0,042-1,404i√3/2=-0,042-0,702i√3

x=(y-b)/3a

Dar ai zis ca te intereseaza doar solutia pozitiva deci o sa inlocuiesc pe y doar cu y₁

x₁=(y₁-b)/3a=(0,084-0)/3*4=0,084/12=0,007

x₁=7/1000