Solve 2a+ax^2+bx+c=x^2+1

$$2 a + a x ^ { 2 } + b x + c = x ^ { 2 } + 1$$

Solve for a

$$a=-\frac{-x^{2}+bx+c-1}{x^{2}+2}$$

Steps for Solving Linear Equation

Subtract bx from both sides.

$$2a+ax^{2}+c=x^{2}+1-bx$$

Subtract c from both sides.

$$2a+ax^{2}=x^{2}+1-bx-c$$

Combine all terms containing a.

$$\left(2+x^{2}\right)a=x^{2}+1-bx-c$$

The equation is in standard form.

$$\left(x^{2}+2\right)a=x^{2}-bx-c+1$$

Divide both sides by 2+x^{2}.

$$\frac{\left(x^{2}+2\right)a}{x^{2}+2}=\frac{x^{2}-bx-c+1}{x^{2}+2}$$

Dividing by 2+x^{2} undoes the multiplication by 2+x^{2}.

$$a=\frac{x^{2}-bx-c+1}{x^{2}+2}$$

Solve for b

$$\left\{\begin{matrix}b=-\frac{ax^{2}-x^{2}+c+2a-1}{x}\text{, }&x\neq 0\\b\in \mathrm{R}\text{, }&a=\frac{1-c}{2}\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract 2a from both sides.

$$ax^{2}+bx+c=x^{2}+1-2a$$

Subtract ax^{2} from both sides.

$$bx+c=x^{2}+1-2a-ax^{2}$$

Subtract c from both sides.

$$bx=x^{2}+1-2a-ax^{2}-c$$

Reorder the terms.

$$bx=-ax^{2}+x^{2}-2a-c+1$$

The equation is in standard form.

$$xb=1-2a-c+x^{2}-ax^{2}$$

Divide both sides by x.

$$\frac{xb}{x}=\frac{1-2a-c+x^{2}-ax^{2}}{x}$$

Dividing by x undoes the multiplication by x.

$$b=\frac{1-2a-c+x^{2}-ax^{2}}{x}$$