Solve for d
Solve for x

## Share

\left(xy^{2}d+xd\right)x+\left(yx^{2}+y\right)dy=0
Use the distributive property to multiply xy^{2}+x by d.
y^{2}dx^{2}+dx^{2}+\left(yx^{2}+y\right)dy=0
Use the distributive property to multiply xy^{2}d+xd by x.
y^{2}dx^{2}+dx^{2}+\left(yx^{2}d+yd\right)y=0
Use the distributive property to multiply yx^{2}+y by d.
y^{2}dx^{2}+dx^{2}+x^{2}dy^{2}+dy^{2}=0
Use the distributive property to multiply yx^{2}d+yd by y.
2y^{2}dx^{2}+dx^{2}+dy^{2}=0
Combine y^{2}dx^{2} and x^{2}dy^{2} to get 2y^{2}dx^{2}.
\left(2y^{2}x^{2}+x^{2}+y^{2}\right)d=0
Combine all terms containing d.
\left(2x^{2}y^{2}+x^{2}+y^{2}\right)d=0
The equation is in standard form.
d=0
Divide 0 by 2y^{2}x^{2}+x^{2}+y^{2}.
\left(xy^{2}d+xd\right)x+\left(yx^{2}+y\right)dy=0
Use the distributive property to multiply xy^{2}+x by d.
y^{2}dx^{2}+dx^{2}+\left(yx^{2}+y\right)dy=0
Use the distributive property to multiply xy^{2}d+xd by x.
y^{2}dx^{2}+dx^{2}+\left(yx^{2}d+yd\right)y=0
Use the distributive property to multiply yx^{2}+y by d.
y^{2}dx^{2}+dx^{2}+x^{2}dy^{2}+dy^{2}=0
Use the distributive property to multiply yx^{2}d+yd by y.
2y^{2}dx^{2}+dx^{2}+dy^{2}=0
Combine y^{2}dx^{2} and x^{2}dy^{2} to get 2y^{2}dx^{2}.
2y^{2}dx^{2}+dx^{2}=-dy^{2}
Subtract dy^{2} from both sides. Anything subtracted from zero gives its negation.
2dx^{2}y^{2}+dx^{2}=-dy^{2}
Reorder the terms.
\left(2dy^{2}+d\right)x^{2}=-dy^{2}
Combine all terms containing x.
\frac{\left(2dy^{2}+d\right)x^{2}}{2dy^{2}+d}=-\frac{dy^{2}}{2dy^{2}+d}
Divide both sides by 2y^{2}d+d.
x^{2}=-\frac{dy^{2}}{2dy^{2}+d}
Dividing by 2y^{2}d+d undoes the multiplication by 2y^{2}d+d.
x^{2}=-\frac{y^{2}}{2y^{2}+1}
Divide -dy^{2} by 2y^{2}d+d.
x=\frac{\sqrt{-y^{2}}}{\sqrt{2y^{2}+1}} x=-\frac{\sqrt{-y^{2}}}{\sqrt{2y^{2}+1}}
Take the square root of both sides of the equation.
\left(xy^{2}d+xd\right)x+\left(yx^{2}+y\right)dy=0
Use the distributive property to multiply xy^{2}+x by d.
y^{2}dx^{2}+dx^{2}+\left(yx^{2}+y\right)dy=0
Use the distributive property to multiply xy^{2}d+xd by x.
y^{2}dx^{2}+dx^{2}+\left(yx^{2}d+yd\right)y=0
Use the distributive property to multiply yx^{2}+y by d.
y^{2}dx^{2}+dx^{2}+x^{2}dy^{2}+dy^{2}=0
Use the distributive property to multiply yx^{2}d+yd by y.
2y^{2}dx^{2}+dx^{2}+dy^{2}=0
Combine y^{2}dx^{2} and x^{2}dy^{2} to get 2y^{2}dx^{2}.
\left(2y^{2}d+d\right)x^{2}+dy^{2}=0
Combine all terms containing x.
\left(2dy^{2}+d\right)x^{2}+dy^{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.
x=\frac{0±\sqrt{0^{2}-4\left(2dy^{2}+d\right)dy^{2}}}{2\left(2dy^{2}+d\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2y^{2}d+d for a, 0 for b, and dy^{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{0±\sqrt{-4\left(2dy^{2}+d\right)dy^{2}}}{2\left(2dy^{2}+d\right)}
Square 0.
x=\frac{0±\sqrt{\left(-8dy^{2}-4d\right)dy^{2}}}{2\left(2dy^{2}+d\right)}
Multiply -4 times 2y^{2}d+d.
x=\frac{0±\sqrt{-4d^{2}y^{2}\left(2y^{2}+1\right)}}{2\left(2dy^{2}+d\right)}
Multiply -8dy^{2}-4d times dy^{2}.
x=\frac{0±2\sqrt{-d^{2}y^{2}\left(2y^{2}+1\right)}}{2\left(2dy^{2}+d\right)}
Take the square root of -4\left(2y^{2}+1\right)y^{2}d^{2}.
x=\frac{0±2\sqrt{-d^{2}y^{2}\left(2y^{2}+1\right)}}{4dy^{2}+2d}
Multiply 2 times 2y^{2}d+d.
x=\frac{|d|\sqrt{-y^{2}}}{d\sqrt{2y^{2}+1}}
Now solve the equation x=\frac{0±2\sqrt{-d^{2}y^{2}\left(2y^{2}+1\right)}}{4dy^{2}+2d} when ± is plus.
x=-\frac{|d|\sqrt{-y^{2}}}{d\sqrt{2y^{2}+1}}
Now solve the equation x=\frac{0±2\sqrt{-d^{2}y^{2}\left(2y^{2}+1\right)}}{4dy^{2}+2d} when ± is minus.
x=\frac{|d|\sqrt{-y^{2}}}{d\sqrt{2y^{2}+1}} x=-\frac{|d|\sqrt{-y^{2}}}{d\sqrt{2y^{2}+1}}
The equation is now solved.