Solve y=3x^2+2ax+7a

$$y = 3 { x }^{ 2 } +2ax+7a$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{3x^{2}-y}{2x+7}\text{, }&x\neq -\frac{7}{2}\\a\in \mathrm{C}\text{, }&y=\frac{147}{4}\text{ and }x=-\frac{7}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$3x^{2}+2ax+7a=y$$

Subtract 3x^{2} from both sides.

$$2ax+7a=y-3x^{2}$$

Combine all terms containing a.

$$\left(2x+7\right)a=y-3x^{2}$$

Divide both sides by 2x+7.

$$\frac{\left(2x+7\right)a}{2x+7}=\frac{y-3x^{2}}{2x+7}$$

Dividing by 2x+7 undoes the multiplication by 2x+7.

$$a=\frac{y-3x^{2}}{2x+7}$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{3x^{2}-y}{2x+7}\text{, }&x\neq -\frac{7}{2}\\a\in \mathrm{R}\text{, }&y=\frac{147}{4}\text{ and }x=-\frac{7}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$3x^{2}+2ax+7a=y$$

Subtract 3x^{2} from both sides.

$$2ax+7a=y-3x^{2}$$

Combine all terms containing a.

$$\left(2x+7\right)a=y-3x^{2}$$

Divide both sides by 2x+7.

$$\frac{\left(2x+7\right)a}{2x+7}=\frac{y-3x^{2}}{2x+7}$$

Dividing by 2x+7 undoes the multiplication by 2x+7.

$$a=\frac{y-3x^{2}}{2x+7}$$

Solve for x (complex solution)

$$x=\frac{\sqrt{3y+a^{2}-21a}-a}{3}$$

$$x=\frac{-\sqrt{3y+a^{2}-21a}-a}{3}$$

Solve for x

$$x=\frac{\sqrt{3y+a^{2}-21a}-a}{3}$$

$$x=\frac{-\sqrt{3y+a^{2}-21a}-a}{3}\text{, }y\geq -\frac{a^{2}}{3}+7a$$