Solve 3(2a-1)x^2-2(a+1)x+1=0

$$3 ( 2 a - 1 ) x ^ { 2 } - 2 ( a + 1 ) x + 1 = 0$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{1}{2}+\frac{1}{2x}\text{, }&x\neq 0\\a\in \mathrm{C}\text{, }&x=\frac{1}{3}\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 3 by 2a-1.

$$\left(6a-3\right)x^{2}-2\left(a+1\right)x+1=0$$

Use the distributive property to multiply 6a-3 by x^{2}.

$$6ax^{2}-3x^{2}-2\left(a+1\right)x+1=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

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

Use the distributive property to multiply -2 by a+1.

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

Use the distributive property to multiply -2a-2 by x.

$$6ax^{2}-3x^{2}-2ax-2x=-1$$

Add 3x^{2} to both sides.

$$6ax^{2}-2ax-2x=-1+3x^{2}$$

Add 2x to both sides.

$$6ax^{2}-2ax=-1+3x^{2}+2x$$

Combine all terms containing a.

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

The equation is in standard form.

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

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

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

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

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

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

$$a=\frac{1}{2}+\frac{1}{2x}$$

Solve for a

$$\left\{\begin{matrix}a=\frac{1}{2}+\frac{1}{2x}\text{, }&x\neq 0\\a\in \mathrm{R}\text{, }&x=\frac{1}{3}\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 3 by 2a-1.

$$\left(6a-3\right)x^{2}-2\left(a+1\right)x+1=0$$

Use the distributive property to multiply 6a-3 by x^{2}.

$$6ax^{2}-3x^{2}-2\left(a+1\right)x+1=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

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

Use the distributive property to multiply -2 by a+1.

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

Use the distributive property to multiply -2a-2 by x.

$$6ax^{2}-3x^{2}-2ax-2x=-1$$

Add 3x^{2} to both sides.

$$6ax^{2}-2ax-2x=-1+3x^{2}$$

Add 2x to both sides.

$$6ax^{2}-2ax=-1+3x^{2}+2x$$

Combine all terms containing a.

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

The equation is in standard form.

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

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

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

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

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

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

$$a=\frac{1}{2}+\frac{1}{2x}$$

Solve for x

$$\left\{\begin{matrix}\\x=\frac{1}{3}\text{, }&\text{unconditionally}\\x=\frac{1}{2a-1}\text{, }&a\neq \frac{1}{2}\end{matrix}\right.$$