Solve 3n^2+5n=n/2[2a+(n-1)d]

$$3 n ^ { 2 } + 5 n = \frac { n } { 2 } [ 2 a + ( n - 1 ) d ]$$

Solve for a (complex solution)

$$\left\{\begin{matrix}\\a=\frac{10+d+6n-dn}{2}\text{, }&\text{unconditionally}\\a\in \mathrm{C}\text{, }&n=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

$$6n^{2}+10n=n\left(2a+\left(n-1\right)d\right)$$

Use the distributive property to multiply n-1 by d.

$$6n^{2}+10n=n\left(2a+nd-d\right)$$

Use the distributive property to multiply n by 2a+nd-d.

$$6n^{2}+10n=2na+dn^{2}-nd$$

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

$$2na+dn^{2}-nd=6n^{2}+10n$$

Subtract dn^{2} from both sides.

$$2na-nd=6n^{2}+10n-dn^{2}$$

Add nd to both sides.

$$2na=6n^{2}+10n-dn^{2}+nd$$

Reorder the terms.

$$2an=-dn^{2}+6n^{2}+dn+10n$$

The equation is in standard form.

$$2na=10n+dn+6n^{2}-dn^{2}$$

Divide both sides by 2n.

$$\frac{2na}{2n}=\frac{n\left(10+d+6n-dn\right)}{2n}$$

Dividing by 2n undoes the multiplication by 2n.

$$a=\frac{n\left(10+d+6n-dn\right)}{2n}$$

Divide n\left(-dn+6n+d+10\right) by 2n.

$$a=-\frac{dn}{2}+\frac{d}{2}+3n+5$$

Solve for d (complex solution)

$$\left\{\begin{matrix}d=-\frac{2\left(-3n+a-5\right)}{n-1}\text{, }&n\neq 1\\d\in \mathrm{C}\text{, }&n=0\text{ or }\left(n=1\text{ and }a=8\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

$$6n^{2}+10n=n\left(2a+\left(n-1\right)d\right)$$

Use the distributive property to multiply n-1 by d.

$$6n^{2}+10n=n\left(2a+nd-d\right)$$

Use the distributive property to multiply n by 2a+nd-d.

$$6n^{2}+10n=2na+dn^{2}-nd$$

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

$$2na+dn^{2}-nd=6n^{2}+10n$$

Subtract 2na from both sides.

$$dn^{2}-nd=6n^{2}+10n-2na$$

Combine all terms containing d.

$$\left(n^{2}-n\right)d=6n^{2}+10n-2na$$

The equation is in standard form.

$$\left(n^{2}-n\right)d=6n^{2}-2an+10n$$

Divide both sides by n^{2}-n.

$$\frac{\left(n^{2}-n\right)d}{n^{2}-n}=\frac{2n\left(3n-a+5\right)}{n^{2}-n}$$

Dividing by n^{2}-n undoes the multiplication by n^{2}-n.

$$d=\frac{2n\left(3n-a+5\right)}{n^{2}-n}$$

Divide 2n\left(5+3n-a\right) by n^{2}-n.

$$d=\frac{2\left(3n-a+5\right)}{n-1}$$

Solve for a

$$\left\{\begin{matrix}\\a=\frac{10+d+6n-dn}{2}\text{, }&\text{unconditionally}\\a\in \mathrm{R}\text{, }&n=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

$$6n^{2}+10n=n\left(2a+\left(n-1\right)d\right)$$

Use the distributive property to multiply n-1 by d.

$$6n^{2}+10n=n\left(2a+nd-d\right)$$

Use the distributive property to multiply n by 2a+nd-d.

$$6n^{2}+10n=2na+dn^{2}-nd$$

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

$$2na+dn^{2}-nd=6n^{2}+10n$$

Subtract dn^{2} from both sides.

$$2na-nd=6n^{2}+10n-dn^{2}$$

Add nd to both sides.

$$2na=6n^{2}+10n-dn^{2}+nd$$

Reorder the terms.

$$2an=-dn^{2}+6n^{2}+dn+10n$$

The equation is in standard form.

$$2na=10n+dn+6n^{2}-dn^{2}$$

Divide both sides by 2n.

$$\frac{2na}{2n}=\frac{n\left(10+d+6n-dn\right)}{2n}$$

Dividing by 2n undoes the multiplication by 2n.

$$a=\frac{n\left(10+d+6n-dn\right)}{2n}$$

Divide n\left(-dn+6n+d+10\right) by 2n.

$$a=-\frac{dn}{2}+\frac{d}{2}+3n+5$$

Solve for d

$$\left\{\begin{matrix}d=-\frac{2\left(-3n+a-5\right)}{n-1}\text{, }&n\neq 1\\d\in \mathrm{R}\text{, }&n=0\text{ or }\left(n=1\text{ and }a=8\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

$$6n^{2}+10n=n\left(2a+\left(n-1\right)d\right)$$

Use the distributive property to multiply n-1 by d.

$$6n^{2}+10n=n\left(2a+nd-d\right)$$

Use the distributive property to multiply n by 2a+nd-d.

$$6n^{2}+10n=2na+dn^{2}-nd$$

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

$$2na+dn^{2}-nd=6n^{2}+10n$$

Subtract 2na from both sides.

$$dn^{2}-nd=6n^{2}+10n-2na$$

Combine all terms containing d.

$$\left(n^{2}-n\right)d=6n^{2}+10n-2na$$

The equation is in standard form.

$$\left(n^{2}-n\right)d=6n^{2}-2an+10n$$

Divide both sides by n^{2}-n.

$$\frac{\left(n^{2}-n\right)d}{n^{2}-n}=\frac{2n\left(3n-a+5\right)}{n^{2}-n}$$

Dividing by n^{2}-n undoes the multiplication by n^{2}-n.

$$d=\frac{2n\left(3n-a+5\right)}{n^{2}-n}$$

Divide 2n\left(5+3n-a\right) by n^{2}-n.

$$d=\frac{2\left(3n-a+5\right)}{n-1}$$