Solve 2an+n^2d+d=8n+6

$$2 a n + n ^ { 2 } d + d = 8 n + 6$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{dn^{2}-8n+d-6}{2n}\text{, }&n\neq 0\\a\in \mathrm{R}\text{, }&d=6\text{ and }n=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract n^{2}d from both sides.

$$2an+d=8n+6-n^{2}d$$

Subtract d from both sides.

$$2an=8n+6-n^{2}d-d$$

Reorder the terms.

$$2an=-dn^{2}+8n-d+6$$

The equation is in standard form.

$$2na=6-d+8n-dn^{2}$$

Divide both sides by 2n.

$$\frac{2na}{2n}=\frac{6-d+8n-dn^{2}}{2n}$$

Dividing by 2n undoes the multiplication by 2n.

$$a=\frac{6-d+8n-dn^{2}}{2n}$$

Divide -dn^{2}+8n-d+6 by 2n.

$$a=-\frac{dn}{2}+\frac{-\frac{d}{2}+3}{n}+4$$

Solve for d

$$d=-\frac{2\left(an-4n-3\right)}{n^{2}+1}$$

Steps for Solving Linear Equation

Subtract 2an from both sides.

$$n^{2}d+d=8n+6-2an$$

Combine all terms containing d.

$$\left(n^{2}+1\right)d=8n+6-2an$$

The equation is in standard form.

$$\left(n^{2}+1\right)d=6+8n-2an$$

Divide both sides by n^{2}+1.

$$\frac{\left(n^{2}+1\right)d}{n^{2}+1}=\frac{6+8n-2an}{n^{2}+1}$$

Dividing by n^{2}+1 undoes the multiplication by n^{2}+1.

$$d=\frac{6+8n-2an}{n^{2}+1}$$

Divide 8n+6-2an by n^{2}+1.

$$d=\frac{2\left(3+4n-an\right)}{n^{2}+1}$$