Solve for x 3/4+1/(2x)=1/(2x)+1/(3x^2)

$\frac{3}{4} + \frac{1}{2 x} = \frac{1}{2 x} + \frac{1}{3 x^{2}}$

Subtract $\frac{3}{4}$ from both sides of the equation.

$\frac{1}{2 x} = \frac{1}{2 x} + \frac{1}{3 x^{2}} - \frac{3}{4}$

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$2 x, 2 x, 3 x^{2}, 4$

Since $2 x, 2 x, 3 x^{2}, 4$ contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $2, 2, 3, 4$ then find LCM for the variable part $x^{1}, x^{1}, x^{2}$ .

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Since $2$ has no factors besides $1$ and $2$ .

$2$ is a prime number

Since $3$ has no factors besides $1$ and $3$ .

$3$ is a prime number

$4$ has factors of $2$ and $2$ .

$2 \cdot 2$

The LCM of $2, 2, 3, 4$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2 \cdot 2 \cdot 3$

The LCM of $2, 2, 3, 4$ is $2 \cdot 2 \cdot 3 = 12$ .

Multiply $2$ by $2$ .

$4 \cdot 3$

Multiply $4$ by $3$ .

$12$

The factor for $x^{1}$ is $x$ itself.

$x^{1} = x$

$x$ occurs $1$ time.

The factors for $x^{2}$ are $x \cdot x$ , which is $x$ multiplied by each other $2$ times.

$x^{2} = x \cdot x$

$x$ occurs $2$ times.

The LCM of $x^{1}, x^{1}, x^{2}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x \cdot x$

Multiply $x$ by $x$ .

$x^{2}$

The LCM for $2 x, 2 x, 3 x^{2}, 4$ is the numeric part $12$ multiplied by the variable part.

$12 x^{2}$

Multiply each term in $\frac{1}{2 x} = \frac{1}{2 x} + \frac{1}{3 x^{2}} - \frac{3}{4}$ by $12 x^{2}$ to eliminate the fractions.

Multiply each term in $\frac{1}{2 x} = \frac{1}{2 x} + \frac{1}{3 x^{2}} - \frac{3}{4}$ by $12 x^{2}$ .

$\frac{1}{2 x} (12 x^{2}) = \frac{1}{2 x} (12 x^{2}) + \frac{1}{3 x^{2}} (12 x^{2}) - \frac{3}{4} (12 x^{2})$

Simplify the left side.

Rewrite using the commutative property of multiplication.

$12 \frac{1}{2 x} x^{2} = \frac{1}{2 x} (12 x^{2}) + \frac{1}{3 x^{2}} (12 x^{2}) - \frac{3}{4} (12 x^{2})$

Cancel the common factor of $2$ .

Factor $2$ out of $12$ .

$2 (6) \frac{1}{2 x} x^{2} = \frac{1}{2 x} (12 x^{2}) + \frac{1}{3 x^{2}} (12 x^{2}) - \frac{3}{4} (12 x^{2})$

Factor $2$ out of $2 x$ .

$2 (6) \frac{1}{2 (x)} x^{2} = \frac{1}{2 x} (12 x^{2}) + \frac{1}{3 x^{2}} (12 x^{2}) - \frac{3}{4} (12 x^{2})$

Cancel the common factor.

$2 \cdot 6 \frac{1}{2 x} x^{2} = \frac{1}{2 x} (12 x^{2}) + \frac{1}{3 x^{2}} (12 x^{2}) - \frac{3}{4} (12 x^{2})$

Rewrite the expression.

$6 \frac{1}{x} x^{2} = \frac{1}{2 x} (12 x^{2}) + \frac{1}{3 x^{2}} (12 x^{2}) - \frac{3}{4} (12 x^{2})$

Combine $6$ and $\frac{1}{x}$ .

$\frac{6}{x} x^{2} = \frac{1}{2 x} (12 x^{2}) + \frac{1}{3 x^{2}} (12 x^{2}) - \frac{3}{4} (12 x^{2})$

Cancel the common factor of $x$ .

Factor $x$ out of $x^{2}$ .

$\frac{6}{x} (x \cdot x) = \frac{1}{2 x} (12 x^{2}) + \frac{1}{3 x^{2}} (12 x^{2}) - \frac{3}{4} (12 x^{2})$

Cancel the common factor.

$\frac{6}{x} (x \cdot x) = \frac{1}{2 x} (12 x^{2}) + \frac{1}{3 x^{2}} (12 x^{2}) - \frac{3}{4} (12 x^{2})$

Rewrite the expression.

$6 x = \frac{1}{2 x} (12 x^{2}) + \frac{1}{3 x^{2}} (12 x^{2}) - \frac{3}{4} (12 x^{2})$

Simplify the right side.

Simplify each term.

Rewrite using the commutative property of multiplication.

$6 x = 12 \frac{1}{2 x} x^{2} + \frac{1}{3 x^{2}} (12 x^{2}) - \frac{3}{4} (12 x^{2})$

Cancel the common factor of $2$ .

Factor $2$ out of $12$ .

$6 x = 2 (6) \frac{1}{2 x} x^{2} + \frac{1}{3 x^{2}} (12 x^{2}) - \frac{3}{4} (12 x^{2})$

Factor $2$ out of $2 x$ .

$6 x = 2 (6) \frac{1}{2 (x)} x^{2} + \frac{1}{3 x^{2}} (12 x^{2}) - \frac{3}{4} (12 x^{2})$

Cancel the common factor.

$6 x = 2 \cdot 6 \frac{1}{2 x} x^{2} + \frac{1}{3 x^{2}} (12 x^{2}) - \frac{3}{4} (12 x^{2})$

Rewrite the expression.

$6 x = 6 \frac{1}{x} x^{2} + \frac{1}{3 x^{2}} (12 x^{2}) - \frac{3}{4} (12 x^{2})$

Combine $6$ and $\frac{1}{x}$ .

$6 x = \frac{6}{x} x^{2} + \frac{1}{3 x^{2}} (12 x^{2}) - \frac{3}{4} (12 x^{2})$

Cancel the common factor of $x$ .

Factor $x$ out of $x^{2}$ .

$6 x = \frac{6}{x} (x \cdot x) + \frac{1}{3 x^{2}} (12 x^{2}) - \frac{3}{4} (12 x^{2})$

Cancel the common factor.

$6 x = \frac{6}{x} (x \cdot x) + \frac{1}{3 x^{2}} (12 x^{2}) - \frac{3}{4} (12 x^{2})$

Rewrite the expression.

$6 x = 6 x + \frac{1}{3 x^{2}} (12 x^{2}) - \frac{3}{4} (12 x^{2})$

Rewrite using the commutative property of multiplication.

$6 x = 6 x + 12 \frac{1}{3 x^{2}} x^{2} - \frac{3}{4} (12 x^{2})$

Cancel the common factor of $3$ .

Factor $3$ out of $12$ .

$6 x = 6 x + 3 (4) \frac{1}{3 x^{2}} x^{2} - \frac{3}{4} (12 x^{2})$

Factor $3$ out of $3 x^{2}$ .

$6 x = 6 x + 3 (4) \frac{1}{3 (x^{2})} x^{2} - \frac{3}{4} (12 x^{2})$

Cancel the common factor.

$6 x = 6 x + 3 \cdot 4 \frac{1}{3 x^{2}} x^{2} - \frac{3}{4} (12 x^{2})$

Rewrite the expression.

$6 x = 6 x + 4 \frac{1}{x^{2}} x^{2} - \frac{3}{4} (12 x^{2})$

Combine $4$ and $\frac{1}{x^{2}}$ .

$6 x = 6 x + \frac{4}{x^{2}} x^{2} - \frac{3}{4} (12 x^{2})$

Cancel the common factor of $x^{2}$ .

Cancel the common factor.

$6 x = 6 x + \frac{4}{x^{2}} x^{2} - \frac{3}{4} (12 x^{2})$

Rewrite the expression.

$6 x = 6 x + 4 - \frac{3}{4} (12 x^{2})$

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$6 x = 6 x + 4 + \frac{- 3}{4} (12 x^{2})$

Factor $4$ out of $12 x^{2}$ .

$6 x = 6 x + 4 + \frac{- 3}{4} (4 (3 x^{2}))$

Cancel the common factor.

$6 x = 6 x + 4 + \frac{- 3}{4} (4 (3 x^{2}))$

Rewrite the expression.

$6 x = 6 x + 4 - 3 (3 x^{2})$

Multiply $3$ by $- 3$ .

$6 x = 6 x + 4 - 9 x^{2}$

Solve the equation.

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$6 x + 4 - 9 x^{2} = 6 x$

Move all terms containing $x$ to the left side of the equation.

Subtract $6 x$ from both sides of the equation.

$6 x + 4 - 9 x^{2} - 6 x = 0$

Combine the opposite terms in $6 x + 4 - 9 x^{2} - 6 x$ .

Subtract $6 x$ from $6 x$ .

$4 - 9 x^{2} + 0 = 0$

Add $4 - 9 x^{2}$ and $0$ .

$4 - 9 x^{2} = 0$

Subtract $4$ from both sides of the equation.

$- 9 x^{2} = - 4$

Divide each term in $- 9 x^{2} = - 4$ by $- 9$ and simplify.

Divide each term in $- 9 x^{2} = - 4$ by $- 9$ .

$\frac{- 9 x^{2}}{- 9} = \frac{- 4}{- 9}$

Simplify the left side.

Cancel the common factor of $- 9$ .

Cancel the common factor.

$\frac{- 9 x^{2}}{- 9} = \frac{- 4}{- 9}$

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 4}{- 9}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x^{2} = \frac{4}{9}$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{4}{9}}$

Simplify $\pm \sqrt{\frac{4}{9}}$ .

Rewrite $\sqrt{\frac{4}{9}}$ as $\frac{\sqrt{4}}{\sqrt{9}}$ .

$x = \pm \frac{\sqrt{4}}{\sqrt{9}}$

Simplify the numerator.

Rewrite $4$ as $2^{2}$ .

$x = \pm \frac{\sqrt{2^{2}}}{\sqrt{9}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm \frac{2}{\sqrt{9}}$

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

$x = \pm \frac{2}{\sqrt{3^{2}}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm \frac{2}{3}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{2}{3}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{2}{3}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{2}{3}, - \frac{2}{3}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2}{3}, - \frac{2}{3}$

Decimal Form:

$x = 0.\overline{6}, - 0.\overline{6}$

Solve for x 3/4+1/(2x)=1/(2x)+1/(3x^2)

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