Solve (x+ax)(x-2ax)+3ax(x+ax)=x^2

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

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&a=-2\text{ or }a=0\end{matrix}\right.$$

Steps by Finding Square Root

Use the distributive property to multiply x+ax by x-2ax and combine like terms.

$$x^{2}-ax^{2}-2a^{2}x^{2}+3ax\left(x+ax\right)=x^{2}$$

Use the distributive property to multiply 3ax by x+ax.

$$x^{2}-ax^{2}-2a^{2}x^{2}+3ax^{2}+3a^{2}x^{2}=x^{2}$$

Combine -ax^{2} and 3ax^{2} to get 2ax^{2}.

$$x^{2}+2ax^{2}-2a^{2}x^{2}+3a^{2}x^{2}=x^{2}$$

Combine -2a^{2}x^{2} and 3a^{2}x^{2} to get a^{2}x^{2}.

$$x^{2}+2ax^{2}+a^{2}x^{2}=x^{2}$$

Subtract x^{2} from both sides.

$$x^{2}+2ax^{2}+a^{2}x^{2}-x^{2}=0$$

Combine x^{2} and -x^{2} to get 0.

$$2ax^{2}+a^{2}x^{2}=0$$

Combine all terms containing x.

$$\left(2a+a^{2}\right)x^{2}=0$$

Dividing by 2a+a^{2} undoes the multiplication by 2a+a^{2}.

$$x^{2}=\frac{0}{a^{2}+2a}$$

Divide 0 by 2a+a^{2}.

$$x^{2}=0$$

Take the square root of both sides of the equation.

$$x=0 x=0$$

The equation is now solved. Solutions are the same.

$$x=0$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x+ax by x-2ax and combine like terms.

$$x^{2}-ax^{2}-2a^{2}x^{2}+3ax\left(x+ax\right)=x^{2}$$

Use the distributive property to multiply 3ax by x+ax.

$$x^{2}-ax^{2}-2a^{2}x^{2}+3ax^{2}+3a^{2}x^{2}=x^{2}$$

Combine -ax^{2} and 3ax^{2} to get 2ax^{2}.

$$x^{2}+2ax^{2}-2a^{2}x^{2}+3a^{2}x^{2}=x^{2}$$

Combine -2a^{2}x^{2} and 3a^{2}x^{2} to get a^{2}x^{2}.

$$x^{2}+2ax^{2}+a^{2}x^{2}=x^{2}$$

Subtract x^{2} from both sides.

$$x^{2}+2ax^{2}+a^{2}x^{2}-x^{2}=0$$

Combine x^{2} and -x^{2} to get 0.

$$2ax^{2}+a^{2}x^{2}=0$$

Combine all terms containing x.

$$\left(2a+a^{2}\right)x^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$\left(a^{2}+2a\right)x^{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2a+a^{2} for a, 0 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}}}{2\left(a^{2}+2a\right)}$$

Take the square root of 0^{2}.

$$x=\frac{0±0}{2\left(a^{2}+2a\right)}$$

Multiply 2 times 2a+a^{2}.

$$x=\frac{0}{2a\left(a+2\right)}$$

Divide 0 by 2a\left(2+a\right).

$$x=0$$

Solve for a (complex solution)

$$\left\{\begin{matrix}\\a=-2\text{; }a=0\text{, }&\text{unconditionally}\\a\in \mathrm{C}\text{, }&x=0\end{matrix}\right.$$

Solve for a

$$\left\{\begin{matrix}\\a=-2\text{; }a=0\text{, }&\text{unconditionally}\\a\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&a=-2\text{ or }a=0\end{matrix}\right.$$