Finite Math Examples

$(\frac{1}{6} x + 6) (\frac{5}{6} x - 1)$

Step 1

Combine fractions.

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Step 1.1

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

$(\frac{x}{6} + 6) (\frac{5}{6} x - 1)$

Step 1.2

Combine $\frac{5}{6}$ and $x$ .

$(\frac{x}{6} + 6) (\frac{5 x}{6} - 1)$

Step 2

Expand $(\frac{x}{6} + 6) (\frac{5 x}{6} - 1)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{x}{6} (\frac{5 x}{6} - 1) + 6 (\frac{5 x}{6} - 1)$

Step 2.2

Apply the distributive property.

$\frac{x}{6} \cdot \frac{5 x}{6} + \frac{x}{6} \cdot - 1 + 6 (\frac{5 x}{6} - 1)$

Step 2.3

Apply the distributive property.

$\frac{x}{6} \cdot \frac{5 x}{6} + \frac{x}{6} \cdot - 1 + 6 \frac{5 x}{6} + 6 \cdot - 1$

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

Multiply $\frac{x}{6} \cdot \frac{5 x}{6}$ .

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Step 3.1.1.1

Multiply $\frac{x}{6}$ by $\frac{5 x}{6}$ .

$\frac{x (5 x)}{6 \cdot 6} + \frac{x}{6} \cdot - 1 + 6 \frac{5 x}{6} + 6 \cdot - 1$

Step 3.1.1.2

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

$\frac{5 (x^{1} x)}{6 \cdot 6} + \frac{x}{6} \cdot - 1 + 6 \frac{5 x}{6} + 6 \cdot - 1$

Step 3.1.1.3

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

$\frac{5 (x^{1} x^{1})}{6 \cdot 6} + \frac{x}{6} \cdot - 1 + 6 \frac{5 x}{6} + 6 \cdot - 1$

Step 3.1.1.4

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

$\frac{5 x^{1 + 1}}{6 \cdot 6} + \frac{x}{6} \cdot - 1 + 6 \frac{5 x}{6} + 6 \cdot - 1$

Step 3.1.1.5

Add $1$ and $1$ .

$\frac{5 x^{2}}{6 \cdot 6} + \frac{x}{6} \cdot - 1 + 6 \frac{5 x}{6} + 6 \cdot - 1$

Step 3.1.1.6

Multiply $6$ by $6$ .

$\frac{5 x^{2}}{36} + \frac{x}{6} \cdot - 1 + 6 \frac{5 x}{6} + 6 \cdot - 1$

Step 3.1.2

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

$\frac{5 x^{2}}{36} + \frac{x \cdot - 1}{6} + 6 \frac{5 x}{6} + 6 \cdot - 1$

Step 3.1.3

Move $- 1$ to the left of $x$ .

$\frac{5 x^{2}}{36} + \frac{- 1 \cdot x}{6} + 6 \frac{5 x}{6} + 6 \cdot - 1$

Step 3.1.4

Move the negative in front of the fraction.

$\frac{5 x^{2}}{36} - \frac{x}{6} + 6 \frac{5 x}{6} + 6 \cdot - 1$

Step 3.1.5

Cancel the common factor of $6$ .

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Step 3.1.5.1

Cancel the common factor.

$\frac{5 x^{2}}{36} - \frac{x}{6} + 6 \frac{5 x}{6} + 6 \cdot - 1$

Step 3.1.5.2

Rewrite the expression.

$\frac{5 x^{2}}{36} - \frac{x}{6} + 5 x + 6 \cdot - 1$

Step 3.1.6

Multiply $6$ by $- 1$ .

$\frac{5 x^{2}}{36} - \frac{x}{6} + 5 x - 6$

Step 3.2

To write $5 x$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{5 x^{2}}{36} - \frac{x}{6} + 5 x \cdot \frac{6}{6} - 6$

Step 3.3

Combine $5 x$ and $\frac{6}{6}$ .

$\frac{5 x^{2}}{36} - \frac{x}{6} + \frac{5 x \cdot 6}{6} - 6$

Step 3.4

Combine the numerators over the common denominator.

$\frac{5 x^{2}}{36} + \frac{- x + 5 x \cdot 6}{6} - 6$

Step 3.5

To write $\frac{- x + 5 x \cdot 6}{6}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{5 x^{2}}{36} + \frac{- x + 5 x \cdot 6}{6} \cdot \frac{6}{6} - 6$

Step 3.6

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.6.1

Multiply $\frac{- x + 5 x \cdot 6}{6}$ by $\frac{6}{6}$ .

$\frac{5 x^{2}}{36} + \frac{(- x + 5 x \cdot 6) \cdot 6}{6 \cdot 6} - 6$

Step 3.6.2

Multiply $6$ by $6$ .

$\frac{5 x^{2}}{36} + \frac{(- x + 5 x \cdot 6) \cdot 6}{36} - 6$

Step 3.7

Combine the numerators over the common denominator.

$\frac{5 x^{2} + (- x + 5 x \cdot 6) \cdot 6}{36} - 6$

Step 3.8

Reorder terms.

$\frac{5 x^{2} + 6 (- x + 5 x \cdot 6)}{36} - 6$

Step 3.9

Combine $\frac{5 x^{2} + 6 (- x + 5 x \cdot 6)}{36}$ and $- 6$ using a common denominator.

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Step 3.9.1

Move $x$ .

$\frac{5 x^{2} + 6 (- x + 5 \cdot 6 x)}{36} - 6$

Step 3.9.2

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{36}{36}$ .

$\frac{5 x^{2} + 6 (- x + 5 \cdot 6 x)}{36} - 6 \cdot \frac{36}{36}$

Step 3.9.3

Combine $- 6$ and $\frac{36}{36}$ .

$\frac{5 x^{2} + 6 (- x + 5 \cdot 6 x)}{36} + \frac{- 6 \cdot 36}{36}$

Step 3.9.4

Combine the numerators over the common denominator.

$\frac{5 x^{2} + 6 (- x + 5 \cdot 6 x) - 6 \cdot 36}{36}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $5$ by $6$ .

$\frac{5 x^{2} + 6 (- x + 30 x) - 6 \cdot 36}{36}$

Step 4.2

Add $- x$ and $30 x$ .

$\frac{5 x^{2} + 6 (29 x) - 6 \cdot 36}{36}$

Step 4.3

Multiply $29$ by $6$ .

$\frac{5 x^{2} + 174 x - 6 \cdot 36}{36}$

Step 4.4

Multiply $- 6$ by $36$ .

$\frac{5 x^{2} + 174 x - 216}{36}$

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 $a \cdot c = 5 \cdot - 216 = - 1080$ and whose sum is $b = 174$ .

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Step 4.5.1.1

Factor $174$ out of $174 x$ .

$\frac{5 x^{2} + 174 (x) - 216}{36}$

Step 4.5.1.2

Rewrite $174$ as $- 6$ plus $180$

$\frac{5 x^{2} + (- 6 + 180) x - 216}{36}$

Step 4.5.1.3

Apply the distributive property.

$\frac{5 x^{2} - 6 x + 180 x - 216}{36}$

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.

$\frac{(5 x^{2} - 6 x) + 180 x - 216}{36}$

Step 4.5.2.2

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

$\frac{x (5 x - 6) + 36 (5 x - 6)}{36}$

Step 4.5.3

Factor the polynomial by factoring out the greatest common factor, $5 x - 6$ .

$\frac{(5 x - 6) (x + 36)}{36}$