Algebra Examples

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

Step 1

Subtract $1$ from both sides of the equation.

$\frac{3}{2 x} - \frac{1}{2 (x + 4)} - 1 = 0$

Step 2

Reorder terms.

$\frac{3}{2 x} - 1 - \frac{1}{2 (x + 4)} = 0$

Step 3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2 x}{2 x}$ .

$\frac{3}{2 x} - 1 \cdot \frac{2 x}{2 x} - \frac{1}{2 (x + 4)} = 0$

Step 4

Combine $- 1$ and $\frac{2 x}{2 x}$ .

$\frac{3}{2 x} + \frac{- (2 x)}{2 x} - \frac{1}{2 (x + 4)} = 0$

Step 5

Combine the numerators over the common denominator.

$\frac{3 - (2 x)}{2 x} - \frac{1}{2 (x + 4)} = 0$

Step 6

Simplify each term.

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Step 6.1

Remove parentheses.

$\frac{3 - 1 \cdot (2 x)}{2 x} - \frac{1}{2 (x + 4)} = 0$

Step 6.2

Multiply $- 1$ by $2$ .

$\frac{3 - 2 x}{2 x} - \frac{1}{2 (x + 4)} = 0$

Step 7

To write $\frac{3 - 2 x}{2 x}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

$\frac{3 - 2 x}{2 x} \cdot \frac{x + 4}{x + 4} - \frac{1}{2 (x + 4)} = 0$

Step 8

To write $- \frac{1}{2 (x + 4)}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{3 - 2 x}{2 x} \cdot \frac{x + 4}{x + 4} - \frac{1}{2 (x + 4)} \cdot \frac{x}{x} = 0$

Step 9

Write each expression with a common denominator of $2 x (x + 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.1

Multiply $\frac{3 - 2 x}{2 x}$ by $\frac{x + 4}{x + 4}$ .

$\frac{(3 - 2 x) (x + 4)}{2 x (x + 4)} - \frac{1}{2 (x + 4)} \cdot \frac{x}{x} = 0$

Step 9.2

Multiply $\frac{1}{2 (x + 4)}$ by $\frac{x}{x}$ .

$\frac{(3 - 2 x) (x + 4)}{2 x (x + 4)} - \frac{x}{2 (x + 4) x} = 0$

Step 9.3

Reorder the factors of $2 (x + 4) x$ .

$\frac{(3 - 2 x) (x + 4)}{2 x (x + 4)} - \frac{x}{2 x (x + 4)} = 0$

Step 10

Combine the numerators over the common denominator.

$\frac{(3 - 2 x) (x + 4) - x}{2 x (x + 4)} = 0$

Step 11

Simplify the numerator.

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Step 11.1

Expand $(3 - 2 x) (x + 4)$ using the FOIL Method.

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Step 11.1.1

Apply the distributive property.

$\frac{3 (x + 4) - 2 x (x + 4) - x}{2 x (x + 4)} = 0$

Step 11.1.2

Apply the distributive property.

$\frac{3 x + 3 \cdot 4 - 2 x (x + 4) - x}{2 x (x + 4)} = 0$

Step 11.1.3

Apply the distributive property.

$\frac{3 x + 3 \cdot 4 - 2 x \cdot x - 2 x \cdot 4 - x}{2 x (x + 4)} = 0$

Step 11.2

Simplify and combine like terms.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Multiply $3$ by $4$ .

$\frac{3 x + 12 - 2 x \cdot x - 2 x \cdot 4 - x}{2 x (x + 4)} = 0$

Step 11.2.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 11.2.1.2.1

Move $x$ .

$\frac{3 x + 12 - 2 (x \cdot x) - 2 x \cdot 4 - x}{2 x (x + 4)} = 0$

Step 11.2.1.2.2

Multiply $x$ by $x$ .

$\frac{3 x + 12 - 2 x^{2} - 2 x \cdot 4 - x}{2 x (x + 4)} = 0$

Step 11.2.1.3

Multiply $4$ by $- 2$ .

$\frac{3 x + 12 - 2 x^{2} - 8 x - x}{2 x (x + 4)} = 0$

Step 11.2.2

Subtract $8 x$ from $3 x$ .

$\frac{- 5 x + 12 - 2 x^{2} - x}{2 x (x + 4)} = 0$

Step 11.3

Subtract $x$ from $- 5 x$ .

$\frac{- 6 x + 12 - 2 x^{2}}{2 x (x + 4)} = 0$

Step 11.4

Reorder terms.

$\frac{- 2 x^{2} - 6 x + 12}{2 x (x + 4)} = 0$

Step 11.5

Factor $2$ out of $- 2 x^{2} - 6 x + 12$ .

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Step 11.5.1

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

$\frac{2 (- x^{2}) - 6 x + 12}{2 x (x + 4)} = 0$

Step 11.5.2

Factor $2$ out of $- 6 x$ .

$\frac{2 (- x^{2}) + 2 (- 3 x) + 12}{2 x (x + 4)} = 0$

Step 11.5.3

Factor $2$ out of $12$ .

$\frac{2 (- x^{2}) + 2 (- 3 x) + 2 (6)}{2 x (x + 4)} = 0$

Step 11.5.4

Factor $2$ out of $2 (- x^{2}) + 2 (- 3 x)$ .

$\frac{2 (- x^{2} - 3 x) + 2 (6)}{2 x (x + 4)} = 0$

Step 11.5.5

Factor $2$ out of $2 (- x^{2} - 3 x) + 2 (6)$ .

$\frac{2 (- x^{2} - 3 x + 6)}{2 x (x + 4)} = 0$

Step 12

Cancel the common factor of $2$ .

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Step 12.1

Cancel the common factor.

$\frac{2 (- x^{2} - 3 x + 6)}{2 x (x + 4)} = 0$

Step 12.2

Rewrite the expression.

$\frac{- x^{2} - 3 x + 6}{x (x + 4)} = 0$

Step 13

Set the numerator equal to zero.

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

Step 14

Solve the equation for $x$ .

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Step 14.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 14.2

Substitute the values $a = - 1$ , $b = - 3$ , and $c = 6$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (- 1 \cdot 6)}}{2 \cdot - 1}$

Step 14.3

Simplify.

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Step 14.3.1

Simplify the numerator.

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Step 14.3.1.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 1 \cdot 6}}{2 \cdot - 1}$

Step 14.3.1.2

Multiply $- 4 \cdot - 1 \cdot 6$ .

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Step 14.3.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{3 \pm \sqrt{9 + 4 \cdot 6}}{2 \cdot - 1}$

Step 14.3.1.2.2

Multiply $4$ by $6$ .

$x = \frac{3 \pm \sqrt{9 + 24}}{2 \cdot - 1}$

Step 14.3.1.3

Add $9$ and $24$ .

$x = \frac{3 \pm \sqrt{33}}{2 \cdot - 1}$

Step 14.3.2

Multiply $2$ by $- 1$ .

$x = \frac{3 \pm \sqrt{33}}{- 2}$

Step 14.3.3

Move the negative in front of the fraction.

$x = - \frac{3 \pm \sqrt{33}}{2}$

Step 14.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 14.4.1

Simplify the numerator.

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Step 14.4.1.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 1 \cdot 6}}{2 \cdot - 1}$

Step 14.4.1.2

Multiply $- 4 \cdot - 1 \cdot 6$ .

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Step 14.4.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{3 \pm \sqrt{9 + 4 \cdot 6}}{2 \cdot - 1}$

Step 14.4.1.2.2

Multiply $4$ by $6$ .

$x = \frac{3 \pm \sqrt{9 + 24}}{2 \cdot - 1}$

Step 14.4.1.3

Add $9$ and $24$ .

$x = \frac{3 \pm \sqrt{33}}{2 \cdot - 1}$

Step 14.4.2

Multiply $2$ by $- 1$ .

$x = \frac{3 \pm \sqrt{33}}{- 2}$

Step 14.4.3

Move the negative in front of the fraction.

$x = - \frac{3 \pm \sqrt{33}}{2}$

Step 14.4.4

Change the $\pm$ to $+$ .

$x = - \frac{3 + \sqrt{33}}{2}$

Step 14.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 14.5.1

Simplify the numerator.

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Step 14.5.1.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 1 \cdot 6}}{2 \cdot - 1}$

Step 14.5.1.2

Multiply $- 4 \cdot - 1 \cdot 6$ .

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Step 14.5.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{3 \pm \sqrt{9 + 4 \cdot 6}}{2 \cdot - 1}$

Step 14.5.1.2.2

Multiply $4$ by $6$ .

$x = \frac{3 \pm \sqrt{9 + 24}}{2 \cdot - 1}$

Step 14.5.1.3

Add $9$ and $24$ .

$x = \frac{3 \pm \sqrt{33}}{2 \cdot - 1}$

Step 14.5.2

Multiply $2$ by $- 1$ .

$x = \frac{3 \pm \sqrt{33}}{- 2}$

Step 14.5.3

Move the negative in front of the fraction.

$x = - \frac{3 \pm \sqrt{33}}{2}$

Step 14.5.4

Change the $\pm$ to $-$ .

$x = - \frac{3 - \sqrt{33}}{2}$

Step 14.6

The final answer is the combination of both solutions.

$x = - \frac{3 + \sqrt{33}}{2}, - \frac{3 - \sqrt{33}}{2}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3 + \sqrt{33}}{2}, - \frac{3 - \sqrt{33}}{2}$

Decimal Form:

$x = - 4.37228132 \dots, 1.37228132 \dots$