Solve 1/2by^2+nA_{s}y-nA_{s}d=0

$$\frac { 1 } { 2 } b y ^ { 2 } + n A _ { s } y - n A _ { s } d = 0$$

Solve for A_s (complex solution)

$$\left\{\begin{matrix}A_{s}=-\frac{by^{2}}{2n\left(y-d\right)}\text{, }&y\neq d\text{ and }n\neq 0\\A_{s}\in \mathrm{C}\text{, }&\left(b=0\text{ and }y=d\right)\text{ or }\left(y=0\text{ and }d=0\right)\text{ or }\left(y=0\text{ and }n=0\text{ and }d\neq 0\right)\text{ or }\left(b=0\text{ and }n=0\text{ and }y\neq d\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract \frac{1}{2}by^{2} from both sides. Anything subtracted from zero gives its negation.

$$nA_{s}y-nA_{s}d=-\frac{1}{2}by^{2}$$

Combine all terms containing A_{s}.

$$\left(ny-nd\right)A_{s}=-\frac{1}{2}by^{2}$$

The equation is in standard form.

$$\left(ny-dn\right)A_{s}=-\frac{by^{2}}{2}$$

Divide both sides by ny-nd.

$$\frac{\left(ny-dn\right)A_{s}}{ny-dn}=-\frac{\frac{by^{2}}{2}}{ny-dn}$$

Dividing by ny-nd undoes the multiplication by ny-nd.

$$A_{s}=-\frac{\frac{by^{2}}{2}}{ny-dn}$$

Divide -\frac{by^{2}}{2} by ny-nd.

$$A_{s}=-\frac{by^{2}}{2n\left(y-d\right)}$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=-\frac{2A_{s}n\left(y-d\right)}{y^{2}}\text{, }&y\neq 0\\b\in \mathrm{C}\text{, }&\left(n=0\text{ or }A_{s}=0\text{ or }d=0\right)\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Add nA_{s}d to both sides.

$$\frac{1}{2}by^{2}+nA_{s}y=0+nA_{s}d$$

Anything plus zero gives itself.

$$\frac{1}{2}by^{2}+nA_{s}y=nA_{s}d$$

Subtract nA_{s}y from both sides.

$$\frac{1}{2}by^{2}=nA_{s}d-nA_{s}y$$

Reorder the terms.

$$\frac{1}{2}by^{2}=-A_{s}ny+A_{s}dn$$

The equation is in standard form.

$$\frac{y^{2}}{2}b=A_{s}dn-A_{s}ny$$

Divide both sides by \frac{1}{2}y^{2}.

$$\frac{2\times \frac{y^{2}}{2}b}{y^{2}}=\frac{2A_{s}n\left(d-y\right)}{y^{2}}$$

Dividing by \frac{1}{2}y^{2} undoes the multiplication by \frac{1}{2}y^{2}.

$$b=\frac{2A_{s}n\left(d-y\right)}{y^{2}}$$

Solve for A_s

$$\left\{\begin{matrix}A_{s}=-\frac{by^{2}}{2n\left(y-d\right)}\text{, }&y\neq d\text{ and }n\neq 0\\A_{s}\in \mathrm{R}\text{, }&\left(b=0\text{ and }y=d\right)\text{ or }\left(y=0\text{ and }d=0\right)\text{ or }\left(y=0\text{ and }n=0\text{ and }d\neq 0\right)\text{ or }\left(b=0\text{ and }n=0\text{ and }y\neq d\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract \frac{1}{2}by^{2} from both sides. Anything subtracted from zero gives its negation.

$$nA_{s}y-nA_{s}d=-\frac{1}{2}by^{2}$$

Combine all terms containing A_{s}.

$$\left(ny-nd\right)A_{s}=-\frac{1}{2}by^{2}$$

The equation is in standard form.

$$\left(ny-dn\right)A_{s}=-\frac{by^{2}}{2}$$

Divide both sides by ny-nd.

$$\frac{\left(ny-dn\right)A_{s}}{ny-dn}=-\frac{\frac{by^{2}}{2}}{ny-dn}$$

Dividing by ny-nd undoes the multiplication by ny-nd.

$$A_{s}=-\frac{\frac{by^{2}}{2}}{ny-dn}$$

Divide -\frac{by^{2}}{2} by ny-nd.

$$A_{s}=-\frac{by^{2}}{2n\left(y-d\right)}$$

Solve for b

$$\left\{\begin{matrix}b=-\frac{2A_{s}n\left(y-d\right)}{y^{2}}\text{, }&y\neq 0\\b\in \mathrm{R}\text{, }&\left(n=0\text{ or }A_{s}=0\text{ or }d=0\right)\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Add nA_{s}d to both sides.

$$\frac{1}{2}by^{2}+nA_{s}y=0+nA_{s}d$$

Anything plus zero gives itself.

$$\frac{1}{2}by^{2}+nA_{s}y=nA_{s}d$$

Subtract nA_{s}y from both sides.

$$\frac{1}{2}by^{2}=nA_{s}d-nA_{s}y$$

Reorder the terms.

$$\frac{1}{2}by^{2}=-A_{s}ny+A_{s}dn$$

The equation is in standard form.

$$\frac{y^{2}}{2}b=A_{s}dn-A_{s}ny$$

Divide both sides by \frac{1}{2}y^{2}.

$$\frac{2\times \frac{y^{2}}{2}b}{y^{2}}=\frac{2A_{s}n\left(d-y\right)}{y^{2}}$$

Dividing by \frac{1}{2}y^{2} undoes the multiplication by \frac{1}{2}y^{2}.

$$b=\frac{2A_{s}n\left(d-y\right)}{y^{2}}$$