Solve y=3(ax^2+x)

$$y = 3 ( a x ^ { 2 } + x )$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{y-3x}{3x^{2}}\text{, }&x\neq 0\\a\in \mathrm{C}\text{, }&y=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 3 by ax^{2}+x.

$$y=3ax^{2}+3x$$

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

$$3ax^{2}+3x=y$$

Subtract 3x from both sides.

$$3ax^{2}=y-3x$$

The equation is in standard form.

$$3x^{2}a=y-3x$$

Divide both sides by 3x^{2}.

$$\frac{3x^{2}a}{3x^{2}}=\frac{y-3x}{3x^{2}}$$

Dividing by 3x^{2} undoes the multiplication by 3x^{2}.

$$a=\frac{y-3x}{3x^{2}}$$

Solve for a

$$\left\{\begin{matrix}a=\frac{y-3x}{3x^{2}}\text{, }&x\neq 0\\a\in \mathrm{R}\text{, }&y=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 3 by ax^{2}+x.

$$y=3ax^{2}+3x$$

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

$$3ax^{2}+3x=y$$

Subtract 3x from both sides.

$$3ax^{2}=y-3x$$

The equation is in standard form.

$$3x^{2}a=y-3x$$

Divide both sides by 3x^{2}.

$$\frac{3x^{2}a}{3x^{2}}=\frac{y-3x}{3x^{2}}$$

Dividing by 3x^{2} undoes the multiplication by 3x^{2}.

$$a=\frac{y-3x}{3x^{2}}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{3\left(4ay+3\right)}-3}{6a}\text{; }x=-\frac{\sqrt{3}\left(\sqrt{4ay+3}+\sqrt{3}\right)}{6a}\text{, }&a\neq 0\\x=\frac{y}{3}\text{, }&a=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{3\left(4ay+3\right)}-3}{6a}\text{; }x=-\frac{\sqrt{3}\left(\sqrt{4ay+3}+\sqrt{3}\right)}{6a}\text{, }&\left(y\leq -\frac{3}{4a}\text{ and }a<0\right)\text{ or }\left(y\geq -\frac{3}{4a}\text{ and }a>0\right)\text{ or }\left(a\neq 0\text{ and }y=-\frac{3}{4a}\right)\\x=\frac{y}{3}\text{, }&a=0\end{matrix}\right.$$