Solve d=sqrt{an-3}

$$d = \sqrt { a n - 3 }$$

Solve for a

$$a=\frac{d^{2}+3}{n}$$

$$d\geq 0\text{ and }n\neq 0$$

Steps for Solving Radical Equations

Swap sides so that all variable terms are on the left hand side.

$$\sqrt{an-3}=d$$

Square both sides of the equation.

$$na-3=d^{2}$$

Add 3 to both sides of the equation.

$$na-3-\left(-3\right)=d^{2}-\left(-3\right)$$

Subtracting -3 from itself leaves 0.

$$na=d^{2}-\left(-3\right)$$

Subtract -3 from d^{2}.

$$na=d^{2}+3$$

Divide both sides by n.

$$\frac{na}{n}=\frac{d^{2}+3}{n}$$

Dividing by n undoes the multiplication by n.

$$a=\frac{d^{2}+3}{n}$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{d^{2}+3}{n}\text{, }&n\neq 0\text{ and }\left(d=0\text{ or }arg(d)<\pi \right)\\a\in \mathrm{C}\text{, }&d=\sqrt{3}i\text{ and }n=0\end{matrix}\right.$$

Solve for d

$$d=\sqrt{an-3}$$

$$\left(n<0\text{ or }a\geq \frac{3}{n}\right)\text{ and }\left(n>0\text{ or }a\leq \frac{3}{n}\right)\text{ and }n\neq 0$$