Question

Calculati

1 + 2x^3 -3x^2=0

Answer 1

$1 + 2x^3 -3x^2=0 \Rightarrow 2x^3-3x^2+1 = 0 \Rightarrow \\ \Rightarrow 2x^3-2x^2-x^2+1 = 0 \Rightarrow 2x^2(x-1)-(x^2-1) = 0 \Rightarrow \\ \Rightarrow 2x^2(x-1)-(x-1)(x+1) = 0 \Rightarrow (x-1)\cdot\Big[2x^2-(x+1)\Big]=0\Rightarrow \\ \Rightarrow (x-1)\cdot(2x^2-x-1) = 0 \\ \\ x_1=1 \\ \\ 2x^2-x-1 = 0 \\ \Delta = 1 +8 = 9 \Rightarrow x_{2,3} = \dfrac{1\pm3}{4} \Rightarrow x_2 = 1,\quad x_3 = -\frac{1}{2} \\ \\ \Rightarrow \boxed{S = \Big\{ -\dfrac{1}{2},1\Big\} }$