Pre-Algebra Examples

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

Step 1

Simplify terms.

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Step 1.1

Combine $\frac{3}{2}$ and $x$ .

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

Step 1.2

Simplify each term.

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Step 1.2.1

Combine $x$ and $\frac{2}{3}$ .

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

Step 1.2.2

Move $2$ to the left of $x$ .

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

Step 2

Expand $(\frac{3 x}{2} - \frac{2}{3}) (- \frac{2 x}{3} - 4)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.1.2

Cancel the common factor of $3$ .

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Step 3.1.2.1

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

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

Step 3.1.2.2

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

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

Step 3.1.2.3

Cancel the common factor.

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

Step 3.1.2.4

Rewrite the expression.

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

Step 3.1.3

Cancel the common factor of $2$ .

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Step 3.1.3.1

Factor $2$ out of $2 x$ .

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

Step 3.1.3.2

Cancel the common factor.

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

Step 3.1.3.3

Rewrite the expression.

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

Step 3.1.4

Raise $x$ to the power of $1$ .

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

Step 3.1.5

Raise $x$ to the power of $1$ .

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

Step 3.1.6

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

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

Step 3.1.7

Add $1$ and $1$ .

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

Step 3.1.8

Cancel the common factor of $2$ .

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Step 3.1.8.1

Factor $2$ out of $- 4$ .

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

Step 3.1.8.2

Cancel the common factor.

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

Step 3.1.8.3

Rewrite the expression.

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

Step 3.1.9

Multiply $- 2$ by $3$ .

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

Step 3.1.10

Multiply $- \frac{2}{3} (- \frac{2 x}{3})$ .

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Step 3.1.10.1

Multiply $- 1$ by $- 1$ .

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

Step 3.1.10.2

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

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

Step 3.1.10.3

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

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

Step 3.1.10.4

Multiply $2$ by $2$ .

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

Step 3.1.10.5

Multiply $3$ by $3$ .

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

Step 3.1.11

Multiply $- \frac{2}{3} \cdot - 4$ .

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Step 3.1.11.1

Multiply $- 4$ by $- 1$ .

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

Step 3.1.11.2

Combine $4$ and $\frac{2}{3}$ .

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

Step 3.1.11.3

Multiply $4$ by $2$ .

$- x^{2} - 6 x + \frac{4 x}{9} + \frac{8}{3}$

Step 3.2

To write $- 6 x$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$- x^{2} - 6 x \cdot \frac{9}{9} + \frac{4 x}{9} + \frac{8}{3}$

Step 3.3

Combine $- 6 x$ and $\frac{9}{9}$ .

$- x^{2} + \frac{- 6 x \cdot 9}{9} + \frac{4 x}{9} + \frac{8}{3}$

Step 3.4

Combine the numerators over the common denominator.

$- x^{2} + \frac{- 6 x \cdot 9 + 4 x}{9} + \frac{8}{3}$

Step 3.5

To write $- x^{2}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$- x^{2} \cdot \frac{9}{9} + \frac{- 6 x \cdot 9 + 4 x}{9} + \frac{8}{3}$

Step 3.6

Combine $- x^{2}$ and $\frac{9}{9}$ .

$\frac{- x^{2} \cdot 9}{9} + \frac{- 6 x \cdot 9 + 4 x}{9} + \frac{8}{3}$

Step 3.7

Combine the numerators over the common denominator.

$\frac{- x^{2} \cdot 9 - 6 x \cdot 9 + 4 x}{9} + \frac{8}{3}$

Step 3.8

To write $\frac{8}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{- x^{2} \cdot 9 - 6 x \cdot 9 + 4 x}{9} + \frac{8}{3} \cdot \frac{3}{3}$

Step 3.9

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.9.1

Multiply $\frac{8}{3}$ by $\frac{3}{3}$ .

$\frac{- x^{2} \cdot 9 - 6 x \cdot 9 + 4 x}{9} + \frac{8 \cdot 3}{3 \cdot 3}$

Step 3.9.2

Multiply $3$ by $3$ .

$\frac{- x^{2} \cdot 9 - 6 x \cdot 9 + 4 x}{9} + \frac{8 \cdot 3}{9}$

Step 3.10

Combine the numerators over the common denominator.

$\frac{- x^{2} \cdot 9 - 6 x \cdot 9 + 4 x + 8 \cdot 3}{9}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $9$ by $- 1$ .

$\frac{- 9 x^{2} - 6 x \cdot 9 + 4 x + 8 \cdot 3}{9}$

Step 4.2

Multiply $9$ by $- 6$ .

$\frac{- 9 x^{2} - 54 x + 4 x + 8 \cdot 3}{9}$

Step 4.3

Multiply $8$ by $3$ .

$\frac{- 9 x^{2} - 54 x + 4 x + 24}{9}$

Step 4.4

Add $- 54 x$ and $4 x$ .

$\frac{- 9 x^{2} - 50 x + 24}{9}$

Step 4.5

Factor by grouping.

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Step 4.5.1

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 $a \cdot c = - 9 \cdot 24 = - 216$ and whose sum is $b = - 50$ .

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Step 4.5.1.1

Factor $- 50$ out of $- 50 x$ .

$\frac{- 9 x^{2} - 50 (x) + 24}{9}$

Step 4.5.1.2

Rewrite $- 50$ as $4$ plus $- 54$

$\frac{- 9 x^{2} + (4 - 54) x + 24}{9}$

Step 4.5.1.3

Apply the distributive property.

$\frac{- 9 x^{2} + 4 x - 54 x + 24}{9}$

Step 4.5.2

Factor out the greatest common factor from each group.

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Step 4.5.2.1

Group the first two terms and the last two terms.

$\frac{(- 9 x^{2} + 4 x) - 54 x + 24}{9}$

Step 4.5.2.2

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

$\frac{x (- 9 x + 4) + 6 (- 9 x + 4)}{9}$

Step 4.5.3

Factor the polynomial by factoring out the greatest common factor, $- 9 x + 4$ .

$\frac{(- 9 x + 4) (x + 6)}{9}$

Step 5

Simplify with factoring out.

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Step 5.1

Factor $- 1$ out of $- 9 x$ .

$\frac{(- (9 x) + 4) (x + 6)}{9}$

Step 5.2

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{(- (9 x) - 1 (- 4)) (x + 6)}{9}$

Step 5.3

Factor $- 1$ out of $- (9 x) - 1 (- 4)$ .

$\frac{- (9 x - 4) (x + 6)}{9}$

Step 5.4

Simplify the expression.

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Step 5.4.1

Rewrite $- (9 x - 4)$ as $- 1 (9 x - 4)$ .

$\frac{- 1 (9 x - 4) (x + 6)}{9}$

Step 5.4.2

Move the negative in front of the fraction.

$- \frac{(9 x - 4) (x + 6)}{9}$