Solve t_{n}=an^2+bn

$$t _ { n } = a n ^ { 2 } + b n$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{bn-t_{n}}{n^{2}}\text{, }&n\neq 0\\a\in \mathrm{R}\text{, }&t_{n}=0\text{ and }n=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$an^{2}+bn=t_{n}$$

Subtract bn from both sides.

$$an^{2}=t_{n}-bn$$

The equation is in standard form.

$$n^{2}a=t_{n}-bn$$

Divide both sides by n^{2}.

$$\frac{n^{2}a}{n^{2}}=\frac{t_{n}-bn}{n^{2}}$$

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

$$a=\frac{t_{n}-bn}{n^{2}}$$

Solve for b

$$\left\{\begin{matrix}b=-an+\frac{t_{n}}{n}\text{, }&n\neq 0\\b\in \mathrm{R}\text{, }&t_{n}=0\text{ and }n=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$an^{2}+bn=t_{n}$$

Subtract an^{2} from both sides.

$$bn=t_{n}-an^{2}$$

Reorder the terms.

$$bn=-an^{2}+t_{n}$$

The equation is in standard form.

$$nb=t_{n}-an^{2}$$

Divide both sides by n.

$$\frac{nb}{n}=\frac{t_{n}-an^{2}}{n}$$

Dividing by n undoes the multiplication by n.

$$b=\frac{t_{n}-an^{2}}{n}$$

Divide t_{n}-an^{2} by n.

$$b=-an+\frac{t_{n}}{n}$$