Solve y=(x-3)^2x_{1}

$$y = ( x - 3 ) ^ { 2 } x _ { 1 }$$

Solve for x_1 (complex solution)

$$\left\{\begin{matrix}x_{1}=\frac{y}{\left(x-3\right)^{2}}\text{, }&x\neq 3\\x_{1}\in \mathrm{C}\text{, }&y=0\text{ and }x=3\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$y=\left(x^{2}-6x+9\right)x_{1}$$

Use the distributive property to multiply x^{2}-6x+9 by x_{1}.

$$y=x^{2}x_{1}-6xx_{1}+9x_{1}$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}x_{1}-6xx_{1}+9x_{1}=y$$

Combine all terms containing x_{1}.

$$\left(x^{2}-6x+9\right)x_{1}=y$$

Divide both sides by x^{2}-6x+9.

$$\frac{\left(x^{2}-6x+9\right)x_{1}}{x^{2}-6x+9}=\frac{y}{x^{2}-6x+9}$$

Dividing by x^{2}-6x+9 undoes the multiplication by x^{2}-6x+9.

$$x_{1}=\frac{y}{x^{2}-6x+9}$$

Divide y by x^{2}-6x+9.

$$x_{1}=\frac{y}{\left(x-3\right)^{2}}$$

Solve for x_1

$$\left\{\begin{matrix}x_{1}=\frac{y}{\left(x-3\right)^{2}}\text{, }&x\neq 3\\x_{1}\in \mathrm{R}\text{, }&y=0\text{ and }x=3\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$y=\left(x^{2}-6x+9\right)x_{1}$$

Use the distributive property to multiply x^{2}-6x+9 by x_{1}.

$$y=x^{2}x_{1}-6xx_{1}+9x_{1}$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}x_{1}-6xx_{1}+9x_{1}=y$$

Combine all terms containing x_{1}.

$$\left(x^{2}-6x+9\right)x_{1}=y$$

Divide both sides by x^{2}-6x+9.

$$\frac{\left(x^{2}-6x+9\right)x_{1}}{x^{2}-6x+9}=\frac{y}{x^{2}-6x+9}$$

Dividing by x^{2}-6x+9 undoes the multiplication by x^{2}-6x+9.

$$x_{1}=\frac{y}{x^{2}-6x+9}$$

Divide y by x^{2}-6x+9.

$$x_{1}=\frac{y}{\left(x-3\right)^{2}}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=3+x_{1}^{-\frac{1}{2}}\sqrt{y}\text{; }x=3-x_{1}^{-\frac{1}{2}}\sqrt{y}\text{, }&x_{1}\neq 0\\x\in \mathrm{C}\text{, }&y=0\text{ and }x_{1}=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=-\sqrt{\frac{y}{x_{1}}}+3\text{; }x=\sqrt{\frac{y}{x_{1}}}+3\text{, }&y\leq 0\text{ and }x_{1}<0\\x=-\sqrt{\frac{y}{x_{1}}}+3\text{; }x=\sqrt{\frac{y}{x_{1}}}+3\text{, }&y\geq 0\text{ and }x_{1}>0\\x\in \mathrm{R}\text{, }&y=0\text{ and }x_{1}=0\end{matrix}\right.$$