\[\frac{3x+2}{x}=\frac{x}{-2x+1}\]

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Examples

Multiply both sides by

x

3x+2=\frac{{x}^{2}}{-2x+1}

Multiply both sides by

-2x+1

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

Expand.

-6{x}^{2}+3x-4x+2={x}^{2}

Simplify

-6{x}^{2}+3x-4x+2

-6{x}^{2}-x+2

-6{x}^{2}-x+2={x}^{2}

Move all terms to one side.

6{x}^{2}+x-2+{x}^{2}=0

Simplify

6{x}^{2}+x-2+{x}^{2}

7{x}^{2}+x-2

7{x}^{2}+x-2=0

Use the Quadratic Formula.

In general, given

a{x}^{2}+bx+c=0

, there exists two solutions where:

x=\frac{-b+\sqrt{{b}^{2}-4ac}}{2a},\frac{-b-\sqrt{{b}^{2}-4ac}}{2a}

In this case,

a=7

b=1

and

c=-2

{x}^{}=\frac{-1+\sqrt{1-4\times 7\times -2}}{2\times 7},\frac{-1-\sqrt{1-4\times 7\times -2}}{2\times 7}

Simplify.

x=\frac{-1+\sqrt{57}}{14},\frac{-1-\sqrt{57}}{14}

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x=\frac{-1+\sqrt{57}}{14},\frac{-1-\sqrt{57}}{14}

Done

Decimal Form: 0.467845, -0.610702