Solve (xy-x^2)dy-y^2dx=.0

$$( x y - x ^ { 2 } ) d y - y ^ { 2 } d x = .0$$

Solve for d (complex solution)

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

Steps for Solving Linear Equation

Use the distributive property to multiply xy-x^{2} by d.

$$\left(xyd-x^{2}d\right)y-y^{2}dx=0$$

Use the distributive property to multiply xyd-x^{2}d by y.

$$xdy^{2}-x^{2}dy-y^{2}dx=0$$

Combine xdy^{2} and -y^{2}dx to get 0.

$$-x^{2}dy=0$$

The equation is in standard form.

$$\left(-yx^{2}\right)d=0$$

Divide 0 by -x^{2}y.

$$d=0$$

Solve for x (complex solution)

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

Steps by Finding Square Root

Use the distributive property to multiply xy-x^{2} by d.

$$\left(xyd-x^{2}d\right)y-y^{2}dx=0$$

Use the distributive property to multiply xyd-x^{2}d by y.

$$xdy^{2}-x^{2}dy-y^{2}dx=0$$

Combine xdy^{2} and -y^{2}dx to get 0.

$$-x^{2}dy=0$$

Dividing by -dy undoes the multiplication by -dy.

$$x^{2}=\frac{0}{-dy}$$

Divide 0 by -dy.

$$x^{2}=0$$

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 xy-x^{2} by d.

$$\left(xyd-x^{2}d\right)y-y^{2}dx=0$$

Use the distributive property to multiply xyd-x^{2}d by y.

$$xdy^{2}-x^{2}dy-y^{2}dx=0$$

Combine xdy^{2} and -y^{2}dx to get 0.

$$-x^{2}dy=0$$

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.

$$\left(-dy\right)x^{2}=0$$

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

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

Take the square root of 0^{2}.

$$x=\frac{0±0}{2\left(-dy\right)}$$

Multiply 2 times -dy.

$$x=\frac{0}{-2dy}$$

Divide 0 by -2dy.

$$x=0$$

Solve for d

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

Steps for Solving Linear Equation

Use the distributive property to multiply xy-x^{2} by d.

$$\left(xyd-x^{2}d\right)y-y^{2}dx=0$$

Use the distributive property to multiply xyd-x^{2}d by y.

$$xdy^{2}-x^{2}dy-y^{2}dx=0$$

Combine xdy^{2} and -y^{2}dx to get 0.

$$-x^{2}dy=0$$

The equation is in standard form.

$$\left(-yx^{2}\right)d=0$$

Divide 0 by -x^{2}y.

$$d=0$$

Solve for x

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