User:A math-wiki/Sandbox

$$ax+b=0$$

$$y+\lambda=x$$

$$a(y+\lambda)+b=0$$

$$ay+b+a\lambda=0$$

$$ax^2+bx+c=0$$

$$y+\lambda=x$$

$$a(y+\lambda)^2+b(y+\lambda)+c=0$$

$$a(y^2+2\lambda y+{\lambda}^2)+b(y+\lambda)+c=0$$

$$ay^2+2a\lambda y+a{\lambda}^2+by+b\lambda+c=0$$

$$ay^2+(b+2a\lambda)y+(c+b\lambda+a{\lambda}^2)=0$$

$$ax^3+bx^2+cx+d=0$$

$$y+\lambda=x$$

$$a(y+\lambda)^3+b(y+\lambda)^2+c(y+\lambda)+d=0$$

$$a(y^3+3\lambda y^2+3{\lambda}^2y+{\lambda}^3)+b(y^2+2\lambda y+{\lambda}^2)+c(y+\lambda)+d=0$$

$$ay^3+3a\lambda y^2+3a{\lambda}^2y+a{\lambda}^3+by^2+2b\lambda y +b{\lambda}^2+cy+c\lambda+d=0$$

$$ay^3+(b+3a\lambda)y^2+(c+2b\lambda+3a{\lambda}^2)y+(d+c\lambda+b{\lambda}^2+a{\lambda}^3)=0$$

$$ax^4+bx^3+cx^2+dx+e=0$$

$$y+\lambda=x$$

$$a(y+\lambda)^4+b(y+\lambda)^3+c(y+\lambda)^2+d(y+\lambda)+e=0$$

$$a(y^4+4\lambda y^3+6{\lambda}^2y^2+4{\lambda}^3y+{\lambda}^4)+b(y^3+3\lambda y^2+3{\lambda}^2y+{\lambda}^3)+c(y^2+2\lambda y+{\lambda}^2)+d(y+\lambda)+e=0$$

$$ay^4+4a\lambda y^3+6a{\lambda}^2y^2+4a{\lambda}^3y+a{\lambda}^4+by^3+3b\lambda y^2+3b{\lambda}^2y+b{\lambda}^3+cy^2+2c\lambda y+c{\lambda}^2+dy+d\lambda+e=0$$

$$ay^4+(b+4a\lambda)y^3+(c+3b\lambda+6a{\lambda}^2)y^2+(d+2c\lambda+3b{\lambda}^2+4a{\lambda}^3)y+(e+d{\lambda}+c{\lambda}^2+b{\lambda}^3+a{\lambda}^4)=0$$

Solutions
$$ax+b=0$$

$$ax=-b$$

$$x=-\frac{b}{a}$$

$$ax^2+bx+c=0$$

$$x^2+\frac{b}{a}x=-\frac{c}{a}$$

$$x^2+\frac{b}{a}x+\frac{b^2}{4a^2}=\frac{b^2}{4a^2}-\frac{4ac}{4a^2}$$

$$(x+\frac{b}{2a})^2=\frac{b^2-4ac}{4a^2}$$

$$\sqrt{(x+\frac{b}{2a})^2}=\pm\sqrt{\frac{b^2-4ac}{4a^2}}$$

$$x+\frac{b}{2a}=\frac{\pm\sqrt{b^2-4ac}}{2a}$$

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

$$ax^3+bx^2+cx+d=0$$

$$x^3+\frac{b}{a}x^2+\frac{c}{a}x+\frac{d}{a}=0$$

$$y-\frac{b}{3a}=x$$

$$(y-\frac{b}{3a})^3+\frac{b}{a}(y-\frac{b}{3a})^2+\frac{c}{a}(y-\frac{b}{3a})+\frac{d}{a}=0$$

$$(y^3-\frac{b}{a}y^2+\frac{b^2}{3a^2}y-\frac{b^3}{27a^3})+\frac{b}{a}(y^2-\frac{2b}{3a}y+\frac{b^2}{9a^2})+\frac{c}{a}(y-\frac{b}{3a})+\frac{d}{a}=0$$

$$y^3-\frac{b}{a}y^2+\frac{b^2}{3a^2}y-\frac{b^3}{27a^3}+\frac{b}{a}y^2-\frac{2b^2}{3a^2}y+\frac{b^3}{9a^3}+\frac{c}{a}y-\frac{bc}{3a^2}+\frac{d}{a}=0$$

$$y^3+(\frac{c}{a}-\frac{b^2}{3a^2})y+(\frac{d}{a}-\frac{bc}{3a^2}+\frac{2b^3}{27a^3})=0$$

$$y^3+(\frac{3ac-b^2}{3a^2})y+(\frac{2b^3-9abc+27a^2d}{27a^3})=0$$

$$y^3+(\frac{3ac-b^2}{3a^2})y=\frac{9abc-2b^3-27a^2d}{27a^3}$$

$$A=\frac{3ac-b^2}{3a^2}$$

$$B=\frac{9abc-2b^3-27a^2d}{27a^3}$$

$$y^3+Ay=B$$

$$y=s-t$$

$$A=3st$$

$$B=s^3-t^3$$

$$(s-t)^3+3st(s-t)=s^3-t^3$$

$$s^3-3s^2t+3st^2-t^3+3s^2t-3st^2=s^3-t^3$$

$$s^3-t^3=s^3-t^3$$

$$s=\frac{A}{3t}$$

$$t=\frac{A}{3s}$$

$$(\frac{A}{3t})^3-t^3=B$$

$$s^3-(\frac{A}{3s})^3=B$$

$$\frac{A^3}{27t^3}-t^3=B$$

$$s^3-\frac{A^3}{27s^3}=B$$

$$\frac{A^3}{27}-t^6=Bt^3$$

$$s^6-\frac{A^3}{27}=Bs^3$$

$$t^6+Bt^3-\frac{A^3}{27}=0$$

$$s^6-Bs^3+\frac{A^3}{27}=0$$

$$t^3=\frac{-B\pm\sqrt{B^2+\frac{4A^3}{27}}}{2}$$

$$s^3=\frac{B\pm\sqrt{B^2-\frac{4A^3}{27}}}{2}$$

$$t=\sqrt[3]{\frac{-B\pm\sqrt{B^2+\frac{4A^3}{27}}}{2}}$$

$$s=\sqrt[3]{\frac{B\pm\sqrt{B^2-\frac{4A^3}{27}}}{2}}$$

$$y=\sqrt[3]{\frac{B\pm\sqrt{B^2-\frac{4A^3}{27}}}{2}}-\sqrt[3]{\frac{-B\pm\sqrt{B^2+\frac{4A^3}{27}}}{2}}$$

$$y=x+\frac{b}{3a}$$

$$x+\frac{b}{3a}=\sqrt[3]{\frac{B\pm\sqrt{B^2-\frac{4A^3}{27}}}{2}}-\sqrt[3]{\frac{-B\pm\sqrt{B^2+\frac{4A^3}{27}}}{2}}$$

$$x=\sqrt[3]{\frac{B\pm\sqrt{B^2-\frac{4A^3}{27}}}{2}}-\sqrt[3]{\frac{-B\pm\sqrt{B^2+\frac{4A^3}{27}}}{2}}-\frac{b}{3a}$$

$$x=\sqrt[3]{\frac{(\frac{9abc-2b^3-27a^2d}{27a^3})\pm\sqrt{(\frac{9abc-2b^3-27a^2d}{27a^3})^2-\frac{4(\frac{3ac-b^2}{3a^2})^3}{27}}}{2}}-\sqrt[3]{\frac{(\frac{2b^3-9abc+27a^2d}{27a^3})\pm\sqrt{(\frac{9abc-2b^3-27a^2d}{27a^3})^2+\frac{4(\frac{3ac-b^2}{3a^2})^3}{27}}}{2}}-\frac{b}{3a}$$

$$x=\sqrt[3]{\frac{(\frac{9abc-2b^3-27a^2d}{27a^3})\pm\sqrt{\frac{(9abc-2b^3-27a^2d)^2}{729a^6}-\frac{4(3ac-b^2)^3}{729a^6}}}{2}}-\sqrt[3]{\frac{(\frac{2b^3-9abc+27a^2d}{27a^3})\pm\sqrt{\frac{(9abc-2b^3-27a^2d)^2}{729a^6}+\frac{4(3ac-b^2)^3}{729a^6}}}{2}}-\frac{b}{3a}$$

$$x=\sqrt[3]{\frac{\frac{9abc-2b^3-27a^2d}{27a^3}\pm\frac{\sqrt{(9abc-2b^3-27a^2d)^2-4(3ac-b^2)^3}}{27a^3}}{2}}-\sqrt[3]{\frac{\frac{2b^3-9abc+27a^2d}{27a^3}\pm\frac{\sqrt{(9abc-2b^3-27a^2d)^2+4(3ac-b^2)^3}}{27a^3}}{2}}-\frac{b}{3a}$$

$$x=\sqrt[3]{\frac{9abc-2b^3-27a^2d\pm\sqrt{(9abc-2b^3-27a^2d)^2-4(3ac-b^2)^3}}{54a^3}}-\sqrt[3]{\frac{2b^3-9abc+27a^2d\pm\sqrt{(9abc-2b^3-27a^2d)^2+4(3ac-b^2)^3}}{54a^3}}-\frac{b}{3a}$$