Solve (dy)^2-2dxdy-3(dx)^2=0

$$( d y ) ^ { 2 } - 2 d x d y - 3 ( d x ) ^ { 2 } = 0$$

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&y=-x\text{ or }y=3x\end{matrix}\right.$$

Steps by Finding Square Root

Multiply d and d to get d^{2}.

$$\left(dy\right)^{2}-2d^{2}xy-3\left(dx\right)^{2}=0$$

Expand \left(dy\right)^{2}.

$$d^{2}y^{2}-2d^{2}xy-3\left(dx\right)^{2}=0$$

Expand \left(dx\right)^{2}.

$$d^{2}y^{2}-2d^{2}xy-3d^{2}x^{2}=0$$

Combine all terms containing d.

$$\left(y^{2}-2xy-3x^{2}\right)d^{2}=0$$

Dividing by y^{2}-2xy-3x^{2} undoes the multiplication by y^{2}-2xy-3x^{2}.

$$d^{2}=\frac{0}{y^{2}-2xy-3x^{2}}$$

Divide 0 by y^{2}-2xy-3x^{2}.

$$d^{2}=0$$

Take the square root of both sides of the equation.

$$d=0 d=0$$

The equation is now solved. Solutions are the same.

$$d=0$$

Steps Using the Quadratic Formula

Multiply d and d to get d^{2}.

$$\left(dy\right)^{2}-2d^{2}xy-3\left(dx\right)^{2}=0$$

Expand \left(dy\right)^{2}.

$$d^{2}y^{2}-2d^{2}xy-3\left(dx\right)^{2}=0$$

Expand \left(dx\right)^{2}.

$$d^{2}y^{2}-2d^{2}xy-3d^{2}x^{2}=0$$

Combine all terms containing d.

$$\left(y^{2}-2xy-3x^{2}\right)d^{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute y^{2}-2xy-3x^{2} for a, 0 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$d=\frac{0±\sqrt{0^{2}}}{2\left(y^{2}-2xy-3x^{2}\right)}$$

Take the square root of 0^{2}.

$$d=\frac{0±0}{2\left(y^{2}-2xy-3x^{2}\right)}$$

Multiply 2 times y^{2}-2xy-3x^{2}.

$$d=\frac{0}{2\left(y-3x\right)\left(x+y\right)}$$

Divide 0 by 2\left(y-3x\right)\left(y+x\right).

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=-y\text{; }x=\frac{y}{3}\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\end{matrix}\right.$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&y=-x\text{ or }y=3x\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x\in \mathrm{R}\text{, }&d=0\\x=-y\text{; }x=\frac{y}{3}\text{, }&d\neq 0\end{matrix}\right.$$