Answer:
The solution of the given equation is [tex]\sqrt[3]{4}-\sqrt[3]{2}[/tex].
Step-by-step explanation:
According to the cardano's method, the solution of the equation is x=u-v. If the equation is
[tex]x^3+px=q[/tex]
Where [tex]u^3-v^3=q[/tex]
[tex]3uv=p[/tex]
The given equation is
[tex]x^3+6x=2[/tex]
Here p=6 and q=2.
[tex]u^3-v^3=2[/tex] .... (1)
[tex]3uv=6[/tex]
[tex]uv=2[/tex]
Taking cube both the sides.
[tex]u^3v^3=8[/tex]
Multiply both sides by 4.
[tex]4u^3v^3=32[/tex] .... (2)
Taking square both the sides of equation (1).
[tex](u^3-v^3)^2=2^2[/tex]
[tex](u^3)^2-2u^3v^3+(v^3)^2=4[/tex] .... (3)
Add equation (2) and (3).
[tex](u^3)^2-2u^3v^3+(v^3)^2+4u^3v^3=4+32[/tex]
[tex](u^3+v^3)^2=36[/tex]
Taking square root both the sides.
[tex]u^3+v^3=6[/tex] .... (4)
On adding equation (1) and (4), we get
[tex]2u^3=8[/tex]
[tex]u^3=4[/tex]
[tex]u=\sqrt[3]{4}[/tex]
On subtracting equation (1) and (4), we get
[tex]-2v^3=-4[/tex]
[tex]v^3=2[/tex]
[tex]v=\sqrt[3]{2}[/tex]
The solution of the equation is
[tex]x=u-v=\sqrt[3]{4}-\sqrt[3]{2}[/tex]
Therefore the solution of the given equation is [tex]\sqrt[3]{4}-\sqrt[3]{2}[/tex].
