Pre-Algebra Examples

$(x - \frac{7}{2}) (x + \frac{2}{7})$

Step 1

Expand $(x - \frac{7}{2}) (x + \frac{2}{7})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$x (x + \frac{2}{7}) - \frac{7}{2} (x + \frac{2}{7})$

Step 1.2

Apply the distributive property.

$x \cdot x + x \frac{2}{7} - \frac{7}{2} (x + \frac{2}{7})$

Step 1.3

Apply the distributive property.

$x \cdot x + x \frac{2}{7} - \frac{7}{2} x - \frac{7}{2} \cdot \frac{2}{7}$

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $x$ by $x$ .

$x^{2} + x \frac{2}{7} - \frac{7}{2} x - \frac{7}{2} \cdot \frac{2}{7}$

Step 2.1.2

Combine $x$ and $\frac{2}{7}$ .

$x^{2} + \frac{x \cdot 2}{7} - \frac{7}{2} x - \frac{7}{2} \cdot \frac{2}{7}$

Step 2.1.3

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

$x^{2} + \frac{2 \cdot x}{7} - \frac{7}{2} x - \frac{7}{2} \cdot \frac{2}{7}$

Step 2.1.4

Combine $x$ and $\frac{7}{2}$ .

$x^{2} + \frac{2 x}{7} - \frac{x \cdot 7}{2} - \frac{7}{2} \cdot \frac{2}{7}$

Step 2.1.5

Move $7$ to the left of $x$ .

$x^{2} + \frac{2 x}{7} - \frac{7 \cdot x}{2} - \frac{7}{2} \cdot \frac{2}{7}$

Step 2.1.6

Cancel the common factor of $7$ .

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Step 2.1.6.1

Move the leading negative in $- \frac{7}{2}$ into the numerator.

$x^{2} + \frac{2 x}{7} - \frac{7 x}{2} + \frac{- 7}{2} \cdot \frac{2}{7}$

Step 2.1.6.2

Factor $7$ out of $- 7$ .

$x^{2} + \frac{2 x}{7} - \frac{7 x}{2} + \frac{7 (- 1)}{2} \cdot \frac{2}{7}$

Step 2.1.6.3

Cancel the common factor.

$x^{2} + \frac{2 x}{7} - \frac{7 x}{2} + \frac{7 \cdot - 1}{2} \cdot \frac{2}{7}$

Step 2.1.6.4

Rewrite the expression.

$x^{2} + \frac{2 x}{7} - \frac{7 x}{2} + \frac{- 1}{2} \cdot 2$

Step 2.1.7

Cancel the common factor of $2$ .

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Step 2.1.7.1

Cancel the common factor.

$x^{2} + \frac{2 x}{7} - \frac{7 x}{2} + \frac{- 1}{2} \cdot 2$

Step 2.1.7.2

Rewrite the expression.

$x^{2} + \frac{2 x}{7} - \frac{7 x}{2} - 1$

Step 2.2

To write $\frac{2 x}{7}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x^{2} + \frac{2 x}{7} \cdot \frac{2}{2} - \frac{7 x}{2} - 1$

Step 2.3

To write $- \frac{7 x}{2}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$x^{2} + \frac{2 x}{7} \cdot \frac{2}{2} - \frac{7 x}{2} \cdot \frac{7}{7} - 1$

Step 2.4

Write each expression with a common denominator of $14$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{2 x}{7}$ by $\frac{2}{2}$ .

$x^{2} + \frac{2 x \cdot 2}{7 \cdot 2} - \frac{7 x}{2} \cdot \frac{7}{7} - 1$

Step 2.4.2

Multiply $7$ by $2$ .

$x^{2} + \frac{2 x \cdot 2}{14} - \frac{7 x}{2} \cdot \frac{7}{7} - 1$

Step 2.4.3

Multiply $\frac{7 x}{2}$ by $\frac{7}{7}$ .

$x^{2} + \frac{2 x \cdot 2}{14} - \frac{7 x \cdot 7}{2 \cdot 7} - 1$

Step 2.4.4

Multiply $2$ by $7$ .

$x^{2} + \frac{2 x \cdot 2}{14} - \frac{7 x \cdot 7}{14} - 1$

Step 2.5

Combine the numerators over the common denominator.

$x^{2} + \frac{2 x \cdot 2 - 7 x \cdot 7}{14} - 1$

Step 2.6

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{14}{14}$ .

$x^{2} \cdot \frac{14}{14} + \frac{2 x \cdot 2 - 7 x \cdot 7}{14} - 1$

Step 2.7

Combine $x^{2}$ and $\frac{14}{14}$ .

$\frac{x^{2} \cdot 14}{14} + \frac{2 x \cdot 2 - 7 x \cdot 7}{14} - 1$

Step 2.8

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 14 + 2 x \cdot 2 - 7 x \cdot 7}{14} - 1$

Step 2.9

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{14}{14}$ .

$\frac{x^{2} \cdot 14 + 2 x \cdot 2 - 7 x \cdot 7}{14} - 1 \cdot \frac{14}{14}$

Step 2.10

Combine $- 1$ and $\frac{14}{14}$ .

$\frac{x^{2} \cdot 14 + 2 x \cdot 2 - 7 x \cdot 7}{14} + \frac{- 1 \cdot 14}{14}$

Step 2.11

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 14 + 2 x \cdot 2 - 7 x \cdot 7 - 1 \cdot 14}{14}$

Step 3

Simplify the numerator.

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Step 3.1

Move $14$ to the left of $x^{2}$ .

$\frac{14 \cdot x^{2} + 2 x \cdot 2 - 7 x \cdot 7 - 1 \cdot 14}{14}$

Step 3.2

Multiply $2$ by $2$ .

$\frac{14 x^{2} + 4 x - 7 x \cdot 7 - 1 \cdot 14}{14}$

Step 3.3

Multiply $7$ by $- 7$ .

$\frac{14 x^{2} + 4 x - 49 x - 1 \cdot 14}{14}$

Step 3.4

Multiply $- 1$ by $14$ .

$\frac{14 x^{2} + 4 x - 49 x - 14}{14}$

Step 3.5

Subtract $49 x$ from $4 x$ .

$\frac{14 x^{2} - 45 x - 14}{14}$

Step 3.6

Factor by grouping.

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Step 3.6.1

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 $a \cdot c = 14 \cdot - 14 = - 196$ and whose sum is $b = - 45$ .

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Step 3.6.1.1

Factor $- 45$ out of $- 45 x$ .

$\frac{14 x^{2} - 45 (x) - 14}{14}$

Step 3.6.1.2

Rewrite $- 45$ as $4$ plus $- 49$

$\frac{14 x^{2} + (4 - 49) x - 14}{14}$

Step 3.6.1.3

Apply the distributive property.

$\frac{14 x^{2} + 4 x - 49 x - 14}{14}$

Step 3.6.2

Factor out the greatest common factor from each group.

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Step 3.6.2.1

Group the first two terms and the last two terms.

$\frac{(14 x^{2} + 4 x) - 49 x - 14}{14}$

Step 3.6.2.2

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

$\frac{2 x (7 x + 2) - 7 (7 x + 2)}{14}$

Step 3.6.3

Factor the polynomial by factoring out the greatest common factor, $7 x + 2$ .

$\frac{(7 x + 2) (2 x - 7)}{14}$