Solve 2n^2+1=O(n^2)

$$2 n ^ { 2 } + 1 = O ( n ^ { 2 } )$$

Solve for O

$$O=2+\frac{1}{n^{2}}$$

$$n\neq 0$$

Steps for Solving Linear Equation

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

$$On^{2}=2n^{2}+1$$

The equation is in standard form.

$$n^{2}O=2n^{2}+1$$

Divide both sides by n^{2}.

$$\frac{n^{2}O}{n^{2}}=\frac{2n^{2}+1}{n^{2}}$$

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

$$O=\frac{2n^{2}+1}{n^{2}}$$

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

$$O=2+\frac{1}{n^{2}}$$

Solve for n (complex solution)

$$n=-i\left(2-O\right)^{-\frac{1}{2}}$$

$$n=i\left(2-O\right)^{-\frac{1}{2}}\text{, }O\neq 2$$

Solve for n

$$n=\sqrt{-\frac{1}{2-O}}$$

$$n=-\sqrt{-\frac{1}{2-O}}\text{, }O>2$$