Algebra Examples

$(1 - \frac{9 x^{2} + 4}{12 x}) \div (\frac{1}{3 x} - \frac{1}{2}) + 1$

Step 1

Simplify terms.

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Step 1.1

Write $1$ as a fraction with a common denominator.

$(\frac{12 x}{12 x} - \frac{9 x^{2} + 4}{12 x}) \div (\frac{1}{3 x} - \frac{1}{2}) + 1$

Step 1.2

Combine the numerators over the common denominator.

$\frac{12 x - (9 x^{2} + 4)}{12 x} \div (\frac{1}{3 x} - \frac{1}{2}) + 1$

Step 1.3

Simplify each term.

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Step 1.3.1

Rewrite the division as a fraction.

$\frac{\frac{12 x - (9 x^{2} + 4)}{12 x}}{\frac{1}{3 x} - \frac{1}{2}} + 1$

Step 1.3.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{12 x - (9 x^{2} + 4)}{12 x} \cdot \frac{1}{\frac{1}{3 x} - \frac{1}{2}} + 1$

Step 1.3.3

Simplify the numerator.

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Step 1.3.3.1

Apply the distributive property.

$\frac{12 x - (9 x^{2}) - 1 \cdot 4}{12 x} \cdot \frac{1}{\frac{1}{3 x} - \frac{1}{2}} + 1$

Step 1.3.3.2

Multiply $9$ by $- 1$ .

$\frac{12 x - 9 x^{2} - 1 \cdot 4}{12 x} \cdot \frac{1}{\frac{1}{3 x} - \frac{1}{2}} + 1$

Step 1.3.3.3

Multiply $- 1$ by $4$ .

$\frac{12 x - 9 x^{2} - 4}{12 x} \cdot \frac{1}{\frac{1}{3 x} - \frac{1}{2}} + 1$

Step 1.3.3.4

Reorder terms.

$\frac{- 9 x^{2} + 12 x - 4}{12 x} \cdot \frac{1}{\frac{1}{3 x} - \frac{1}{2}} + 1$

Step 1.3.3.5

Factor by grouping.

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Step 1.3.3.5.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 9 \cdot - 4 = 36$ and whose sum is $b = 12$ .

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Step 1.3.3.5.1.1

Factor $12$ out of $12 x$ .

$\frac{- 9 x^{2} + 12 (x) - 4}{12 x} \cdot \frac{1}{\frac{1}{3 x} - \frac{1}{2}} + 1$

Step 1.3.3.5.1.2

Rewrite $12$ as $6$ plus $6$

$\frac{- 9 x^{2} + (6 + 6) x - 4}{12 x} \cdot \frac{1}{\frac{1}{3 x} - \frac{1}{2}} + 1$

Step 1.3.3.5.1.3

Apply the distributive property.

$\frac{- 9 x^{2} + 6 x + 6 x - 4}{12 x} \cdot \frac{1}{\frac{1}{3 x} - \frac{1}{2}} + 1$

Step 1.3.3.5.2

Factor out the greatest common factor from each group.

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Step 1.3.3.5.2.1

Group the first two terms and the last two terms.

$\frac{(- 9 x^{2} + 6 x) + 6 x - 4}{12 x} \cdot \frac{1}{\frac{1}{3 x} - \frac{1}{2}} + 1$

Step 1.3.3.5.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{3 x (- 3 x + 2) - 2 (- 3 x + 2)}{12 x} \cdot \frac{1}{\frac{1}{3 x} - \frac{1}{2}} + 1$

Step 1.3.3.5.3

Factor the polynomial by factoring out the greatest common factor, $- 3 x + 2$ .

$\frac{(- 3 x + 2) (3 x - 2)}{12 x} \cdot \frac{1}{\frac{1}{3 x} - \frac{1}{2}} + 1$

Step 1.3.4

Simplify the numerator.

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Step 1.3.4.1

Factor $- 1$ out of $- 3 x$ .

$\frac{(- (3 x) + 2) (3 x - 2)}{12 x} \cdot \frac{1}{\frac{1}{3 x} - \frac{1}{2}} + 1$

Step 1.3.4.2

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{(- (3 x) - 1 (- 2)) (3 x - 2)}{12 x} \cdot \frac{1}{\frac{1}{3 x} - \frac{1}{2}} + 1$

Step 1.3.4.3

Factor $- 1$ out of $- (3 x) - 1 (- 2)$ .

$\frac{- (3 x - 2) (3 x - 2)}{12 x} \cdot \frac{1}{\frac{1}{3 x} - \frac{1}{2}} + 1$

Step 1.3.4.4

Rewrite $- (3 x - 2)$ as $- 1 (3 x - 2)$ .

$\frac{- 1 (3 x - 2) (3 x - 2)}{12 x} \cdot \frac{1}{\frac{1}{3 x} - \frac{1}{2}} + 1$

Step 1.3.4.5

Raise $3 x - 2$ to the power of $1$ .

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

Step 1.3.4.6

Raise $3 x - 2$ to the power of $1$ .

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

Step 1.3.4.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.3.4.8

Add $1$ and $1$ .

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

Step 1.3.5

Move the negative in front of the fraction.

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

Step 1.3.6

Simplify the denominator.

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Step 1.3.6.1

To write $\frac{1}{3 x}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.3.6.2

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3 x}{3 x}$ .

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

Step 1.3.6.3

Write each expression with a common denominator of $6 x$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.6.3.1

Multiply $\frac{1}{3 x}$ by $\frac{2}{2}$ .

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

Step 1.3.6.3.2

Multiply $2$ by $3$ .

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

Step 1.3.6.3.3

Multiply $\frac{1}{2}$ by $\frac{3 x}{3 x}$ .

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

Step 1.3.6.3.4

Multiply $3$ by $2$ .

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

Step 1.3.6.4

Combine the numerators over the common denominator.

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

Step 1.3.6.5

Reorder terms.

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

Step 1.3.7

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.3.8

Multiply $\frac{6 x}{- 3 x + 2}$ by $1$ .

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

Step 1.3.9

Cancel the common factor of $6 x$ .

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Step 1.3.9.1

Move the leading negative in $- \frac{{(3 x - 2)}^{2}}{12 x}$ into the numerator.

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

Step 1.3.9.2

Factor $6 x$ out of $12 x$ .

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

Step 1.3.9.3

Cancel the common factor.

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

Step 1.3.9.4

Rewrite the expression.

$\frac{- {(3 x - 2)}^{2}}{2} \cdot \frac{1}{- 3 x + 2} + 1$

Step 1.3.10

Multiply $\frac{- {(3 x - 2)}^{2}}{2}$ by $\frac{1}{- 3 x + 2}$ .

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

Step 1.3.11

Cancel the common factor of ${(3 x - 2)}^{2}$ and $- 3 x + 2$ .

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Step 1.3.11.1

Factor $- 1$ out of $3 x$ .

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

Step 1.3.11.2

Rewrite $- 2$ as $- 1 (2)$ .

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

Step 1.3.11.3

Factor $- 1$ out of $- (- 3 x) - 1 (2)$ .

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

Step 1.3.11.4

Rewrite $- (- 3 x + 2)$ as $- 1 (- 3 x + 2)$ .

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

Step 1.3.11.5

Apply the product rule to $- 1 (- 3 x + 2)$ .

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

Step 1.3.11.6

Raise $- 1$ to the power of $2$ .

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

Step 1.3.11.7

Multiply ${(- 3 x + 2)}^{2}$ by $1$ .

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

Step 1.3.11.8

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

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

Step 1.3.11.9

Cancel the common factors.

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Step 1.3.11.9.1

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

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

Step 1.3.11.9.2

Cancel the common factor.

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

Step 1.3.11.9.3

Rewrite the expression.

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

Step 1.3.12

Move the negative in front of the fraction.

$- \frac{- 3 x + 2}{2} + 1$

Step 1.4

Combine into one fraction.

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Step 1.4.1

Write $1$ as a fraction with a common denominator.

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

Step 1.4.2

Combine the numerators over the common denominator.

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

Step 2

Simplify the numerator.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Multiply $- 3$ by $- 1$ .

$\frac{3 x - 1 \cdot 2 + 2}{2}$

Step 2.3

Multiply $- 1$ by $2$ .

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

Step 2.4

Add $- 2$ and $2$ .

$\frac{3 x + 0}{2}$

Step 2.5

Add $3 x$ and $0$ .

$\frac{3 x}{2}$