Solve x_1^3-x_0^3-(x_1^2-x_0^2)+a(x_{1}-x_{0})=0

$$x _ { 1 } ^ { 3 } - x _ { 0 } ^ { 3 } - ( x _ { 1 } ^ { 2 } - x _ { 0 } ^ { 2 } ) + a ( x _ { 1 } - x _ { 0 } ) = 0$$

Solve for a (complex solution)

$$\left\{\begin{matrix}\\a=x_{1}-x_{1}^{2}+x_{0}-x_{0}x_{1}-x_{0}^{2}\text{, }&\text{unconditionally}\\a\in \mathrm{C}\text{, }&x_{1}=x_{0}\end{matrix}\right.$$

Steps for Solving Linear Equation

To find the opposite of x_{1}^{2}-x_{0}^{2}, find the opposite of each term.

$$x_{1}^{3}-x_{0}^{3}-x_{1}^{2}+x_{0}^{2}+a\left(x_{1}-x_{0}\right)=0$$

Use the distributive property to multiply a by x_{1}-x_{0}.

$$x_{1}^{3}-x_{0}^{3}-x_{1}^{2}+x_{0}^{2}+ax_{1}-ax_{0}=0$$

Subtract x_{1}^{3} from both sides. Anything subtracted from zero gives its negation.

$$-x_{0}^{3}-x_{1}^{2}+x_{0}^{2}+ax_{1}-ax_{0}=-x_{1}^{3}$$

Add x_{0}^{3} to both sides.

$$-x_{1}^{2}+x_{0}^{2}+ax_{1}-ax_{0}=-x_{1}^{3}+x_{0}^{3}$$

Add x_{1}^{2} to both sides.

$$x_{0}^{2}+ax_{1}-ax_{0}=-x_{1}^{3}+x_{0}^{3}+x_{1}^{2}$$

Subtract x_{0}^{2} from both sides.

$$ax_{1}-ax_{0}=-x_{1}^{3}+x_{0}^{3}+x_{1}^{2}-x_{0}^{2}$$

Combine all terms containing a.

$$\left(x_{1}-x_{0}\right)a=-x_{1}^{3}+x_{0}^{3}+x_{1}^{2}-x_{0}^{2}$$

The equation is in standard form.

$$\left(x_{1}-x_{0}\right)a=x_{0}^{3}-x_{0}^{2}-x_{1}^{3}+x_{1}^{2}$$

Divide both sides by x_{1}-x_{0}.

$$\frac{\left(x_{1}-x_{0}\right)a}{x_{1}-x_{0}}=\frac{x_{0}^{3}-x_{0}^{2}-x_{1}^{3}+x_{1}^{2}}{x_{1}-x_{0}}$$

Dividing by x_{1}-x_{0} undoes the multiplication by x_{1}-x_{0}.

$$a=\frac{x_{0}^{3}-x_{0}^{2}-x_{1}^{3}+x_{1}^{2}}{x_{1}-x_{0}}$$

Divide -x_{0}^{2}+x_{0}^{3}+x_{1}^{2}-x_{1}^{3} by x_{1}-x_{0}.

$$a=x_{1}-x_{1}^{2}+x_{0}-x_{0}x_{1}-x_{0}^{2}$$

Solve for a

$$\left\{\begin{matrix}\\a=x_{1}-x_{1}^{2}+x_{0}-x_{0}x_{1}-x_{0}^{2}\text{, }&\text{unconditionally}\\a\in \mathrm{R}\text{, }&x_{1}=x_{0}\end{matrix}\right.$$

Steps for Solving Linear Equation

To find the opposite of x_{1}^{2}-x_{0}^{2}, find the opposite of each term.

$$x_{1}^{3}-x_{0}^{3}-x_{1}^{2}+x_{0}^{2}+a\left(x_{1}-x_{0}\right)=0$$

Use the distributive property to multiply a by x_{1}-x_{0}.

$$x_{1}^{3}-x_{0}^{3}-x_{1}^{2}+x_{0}^{2}+ax_{1}-ax_{0}=0$$

Subtract x_{1}^{3} from both sides. Anything subtracted from zero gives its negation.

$$-x_{0}^{3}-x_{1}^{2}+x_{0}^{2}+ax_{1}-ax_{0}=-x_{1}^{3}$$

Add x_{0}^{3} to both sides.

$$-x_{1}^{2}+x_{0}^{2}+ax_{1}-ax_{0}=-x_{1}^{3}+x_{0}^{3}$$

Add x_{1}^{2} to both sides.

$$x_{0}^{2}+ax_{1}-ax_{0}=-x_{1}^{3}+x_{0}^{3}+x_{1}^{2}$$

Subtract x_{0}^{2} from both sides.

$$ax_{1}-ax_{0}=-x_{1}^{3}+x_{0}^{3}+x_{1}^{2}-x_{0}^{2}$$

Combine all terms containing a.

$$\left(x_{1}-x_{0}\right)a=-x_{1}^{3}+x_{0}^{3}+x_{1}^{2}-x_{0}^{2}$$

The equation is in standard form.

$$\left(x_{1}-x_{0}\right)a=x_{0}^{3}-x_{0}^{2}-x_{1}^{3}+x_{1}^{2}$$

Divide both sides by x_{1}-x_{0}.

$$\frac{\left(x_{1}-x_{0}\right)a}{x_{1}-x_{0}}=\frac{x_{0}^{3}-x_{0}^{2}-x_{1}^{3}+x_{1}^{2}}{x_{1}-x_{0}}$$

Dividing by x_{1}-x_{0} undoes the multiplication by x_{1}-x_{0}.

$$a=\frac{x_{0}^{3}-x_{0}^{2}-x_{1}^{3}+x_{1}^{2}}{x_{1}-x_{0}}$$

Divide -x_{0}^{2}+x_{0}^{3}+x_{1}^{2}-x_{1}^{3} by x_{1}-x_{0}.

$$a=x_{1}-x_{1}^{2}+x_{0}-x_{0}x_{1}-x_{0}^{2}$$

Solve for x_0 (complex solution)

$$x_{0}=\frac{\sqrt{1-4a+2x_{1}-3x_{1}^{2}}-x_{1}+1}{2}$$

$$x_{0}=\frac{-\sqrt{1-4a+2x_{1}-3x_{1}^{2}}-x_{1}+1}{2}$$

$$x_{0}=x_{1}$$

Solve for x_0

$$\left\{\begin{matrix}\\x_{0}=x_{1}\text{, }&\text{unconditionally}\\x_{0}=\frac{\sqrt{1-4a+2x_{1}-3x_{1}^{2}}-x_{1}+1}{2}\text{; }x_{0}=\frac{-\sqrt{1-4a+2x_{1}-3x_{1}^{2}}-x_{1}+1}{2}\text{, }&a\leq -\frac{3x_{1}^{2}}{4}+\frac{x_{1}}{2}+\frac{1}{4}\end{matrix}\right.$$