a)
[tex]x^3-13x-12=0\\\\x^3-x-12x-12=0\\\\x(x^2-1)-12(x+1)=0\\\\x(x-1)(x+1)-12(x+1)=0\\\\(x+1)[x(x-1)-12]=0\\\\x+1=0\quad\vee\quad x(x-1)-12=0\\\\x=-1\qquad\vee\quad x^2-x-12=0\\\\{}\qquad\qquad\qquad x^2-4x+3x-12=0\\\\{}\qquad\qquad\qquad x(x-4)+3(x-4)=0\\\\{}\qquad\qquad\qquad (x-4)(x+3)=0\\\\{}\qquad\qquad\qquad x-4=0\quad\vee\quad x+3=0\\\\{}\qquad\qquad\qquad\ x=4\qquad\vee\quad x=-3\\\\x\in\{-3,\ -1,\ 4\}[/tex]
b)
[tex]x-3+2x(x^2-9)=0\\\\x-3+2x(x-3)(x+3)=0\\\\(x-3)[1+2x(x+3)]=0\\\\(x-3)[1+2x^2+6x]=0\\\\x-3=0\quad\vee\quad2x^2+6x+1=0\\{}\qquad\qquad\qquad\ \Delta=6^2-4\cdot2\cdot1=28\ \implies \sqrt\Delta=2\sqrt7\\\\x=3\qquad\vee\quad x = \frac{-6-2\sqrt7}{2\cdot2}= -\frac{3+\sqrt7}2\qquad\vee\quad x=-\frac{3-\sqrt7}2\\\\x\in\{-\frac{3+\sqrt7}2\,;\ -\frac{3-\sqrt7}2\,;\ 3\}[/tex]
c)
[tex](x^3+1)^2=81x^6\\\\(x^3+1)^2-(9x^3)^2=0\\\\(x^3+1-9x^3)(x^3+1+9x^3)=0\\\\(1-8x^3)(1+10x^3)=0\\\\1-8x^3=0\qquad\vee\qquad1+10x^3=0\\\\ 8x^3= 1\quad\qquad\vee\qquad\ 10x^3=-1\\\\ x^3=\frac18\quad\qquad\vee\qquad\ x^3 =-\frac1{10}\\\\ x=\frac12\quad\qquad\vee\qquad\ x =-\frac1{\sqrt[3]{10}} =-\frac{\sqrt[3]{100}}{10}\\\\x\in\left\{-\frac{\sqrt[3]{100}}{10}\,;\ \frac12\right\}[/tex]
d)
[tex]x^5-2x^3=(2-x^2)x^2\\\\x^3(x^2-2)-(2-x^2)x^2=0\\\\x^3(x^2-2)+(x^2-2)x^2=0\\\\(x^2-2)(x^3-x^2)=0\\\\x^2(x-\sqrt2)(x+\sqrt2)(x-1)=0\\\\x^2=0\ \vee\ x-\sqrt2=0\ \vee\ x+\sqrt2=0\ \vee\ x-1=0\\\\{}\ x=0\ \ \vee\ \ x=\sqrt2\quad \vee\quad x=-\sqrt2\quad \vee\quad x=1\\\\x\in\{-\sqrt{2}\,,\ 0\,,\ 1,\ \sqrt2\}[/tex]
