Solve y=ax^2+x

$$y = a x ^ { 2 } + x$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{x-y}{x^{2}}\text{, }&x\neq 0\\a\in \mathrm{C}\text{, }&y=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$ax^{2}+x=y$$

Subtract x from both sides.

$$ax^{2}=y-x$$

The equation is in standard form.

$$x^{2}a=y-x$$

Divide both sides by x^{2}.

$$\frac{x^{2}a}{x^{2}}=\frac{y-x}{x^{2}}$$

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

$$a=\frac{y-x}{x^{2}}$$

Solve for a

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

Steps for Solving Linear Equation

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

$$ax^{2}+x=y$$

Subtract x from both sides.

$$ax^{2}=y-x$$

The equation is in standard form.

$$x^{2}a=y-x$$

Divide both sides by x^{2}.

$$\frac{x^{2}a}{x^{2}}=\frac{y-x}{x^{2}}$$

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

$$a=\frac{y-x}{x^{2}}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{\sqrt{4ay+1}+1}{2a}\text{; }x=-\frac{-\sqrt{4ay+1}+1}{2a}\text{, }&a\neq 0\\x=y\text{, }&a=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{\sqrt{4ay+1}+1}{2a}\text{; }x=-\frac{-\sqrt{4ay+1}+1}{2a}\text{, }&\left(y\leq -\frac{1}{4a}\text{ and }a<0\right)\text{ or }\left(y\geq -\frac{1}{4a}\text{ and }a>0\right)\text{ or }\left(a\neq 0\text{ and }y=-\frac{1}{4a}\right)\\x=y\text{, }&a=0\end{matrix}\right.$$