Question

Bunăă! Care sunt cele 3 soluții ale ecuației : 2x^3 + 3x^2 + 3x +1 =0?

Answer 1

 

$\displaystyle\bf\\2x^3+3x^2+3x+1=0\\\\Descompunem:\\\\3x^2=x^2+2x^2\\\\3x=x+2x\\\\Rezulta:\\2x^3+(x^2+2x^2)+(x+2x)+1=0\\\\(2x^3+x^2)+(2x^2+x)+(2x+1)=0\\\\x^2(2x+1)+x(2x+1)+(2x+1)=0\\\\(2x+1)(x^2+x+1)=0\\\\2x+1=0~\implies~\boxed{\bf x_1=-\frac{1}{2}}\\\\x^2+x+1=0\\\\x_{23}=\frac{-1\pm\sqrt{1-4}}{2}=\frac{-1\pm\sqrt{-3}}{2}=\frac{-1\pm i\sqrt{3}}{2}=-\frac{1}{2}\pm\frac{i\sqrt{3}}{2}\\\\\boxed{\bf x_2=-\frac{1}{2}+\frac{\sqrt{3}}{2}i}\\\\\boxed{\bf x_3=-\frac{1}{2}-\frac{\sqrt{3}}{2}i}$