Solve x:x^2+(3-2a)x-6a=0

$$x : x ^ { 2 } + ( 3 - 2 a ) x - 6 a = 0$$

Solve for a

$$a=\frac{3x^{2}+1}{2x\left(x+3\right)}$$

$$x\neq -3\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x^{2}.

$$x+\left(3-2a\right)xx^{2}-6ax^{2}=0$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$x+\left(3-2a\right)x^{3}-6ax^{2}=0$$

Use the distributive property to multiply 3-2a by x^{3}.

$$x+3x^{3}-2ax^{3}-6ax^{2}=0$$

Subtract x from both sides. Anything subtracted from zero gives its negation.

$$3x^{3}-2ax^{3}-6ax^{2}=-x$$

Subtract 3x^{3} from both sides.

$$-2ax^{3}-6ax^{2}=-x-3x^{3}$$

Combine all terms containing a.

$$\left(-2x^{3}-6x^{2}\right)a=-x-3x^{3}$$

The equation is in standard form.

$$\left(-2x^{3}-6x^{2}\right)a=-3x^{3}-x$$

Divide both sides by -2x^{3}-6x^{2}.

$$\frac{\left(-2x^{3}-6x^{2}\right)a}{-2x^{3}-6x^{2}}=-\frac{x\left(3x^{2}+1\right)}{-2x^{3}-6x^{2}}$$

Dividing by -2x^{3}-6x^{2} undoes the multiplication by -2x^{3}-6x^{2}.

$$a=-\frac{x\left(3x^{2}+1\right)}{-2x^{3}-6x^{2}}$$

Divide -x\left(1+3x^{2}\right) by -2x^{3}-6x^{2}.

$$a=\frac{3x^{2}+1}{2x\left(x+3\right)}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{9a^{2}+2a-3}+3a}{3-2a}\text{; }x=\frac{-\sqrt{9a^{2}+2a-3}+3a}{3-2a}\text{, }&a\neq \frac{3}{2}\\x=\frac{1}{9}\text{, }&a=\frac{3}{2}\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{9a^{2}+2a-3}+3a}{3-2a}\text{; }x=\frac{-\sqrt{9a^{2}+2a-3}+3a}{3-2a}\text{, }&a\leq \frac{-2\sqrt{7}-1}{9}\text{ or }\left(a\neq \frac{3}{2}\text{ and }a\geq \frac{2\sqrt{7}-1}{9}\right)\\x=\frac{1}{9}\text{, }&a=\frac{3}{2}\end{matrix}\right.$$