Finite Math Examples

$(\frac{3}{4} r - \frac{2}{3} s) (\frac{5}{4} r + \frac{1}{3} s)$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$(\frac{3 r}{4} - \frac{2}{3} s) (\frac{5}{4} r + \frac{1}{3} s)$

Step 1.1.2

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

$(\frac{3 r}{4} - \frac{s \cdot 2}{3}) (\frac{5}{4} r + \frac{1}{3} s)$

Step 1.1.3

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

$(\frac{3 r}{4} - \frac{2 s}{3}) (\frac{5}{4} r + \frac{1}{3} s)$

Step 1.2

Simplify each term.

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Step 1.2.1

Combine $\frac{5}{4}$ and $r$ .

$(\frac{3 r}{4} - \frac{2 s}{3}) (\frac{5 r}{4} + \frac{1}{3} s)$

Step 1.2.2

Combine $\frac{1}{3}$ and $s$ .

$(\frac{3 r}{4} - \frac{2 s}{3}) (\frac{5 r}{4} + \frac{s}{3})$

Step 2

Expand $(\frac{3 r}{4} - \frac{2 s}{3}) (\frac{5 r}{4} + \frac{s}{3})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{3 r}{4} (\frac{5 r}{4} + \frac{s}{3}) - \frac{2 s}{3} (\frac{5 r}{4} + \frac{s}{3})$

Step 2.2

Apply the distributive property.

$\frac{3 r}{4} \cdot \frac{5 r}{4} + \frac{3 r}{4} \cdot \frac{s}{3} - \frac{2 s}{3} (\frac{5 r}{4} + \frac{s}{3})$

Step 2.3

Apply the distributive property.

$\frac{3 r}{4} \cdot \frac{5 r}{4} + \frac{3 r}{4} \cdot \frac{s}{3} - \frac{2 s}{3} \cdot \frac{5 r}{4} - \frac{2 s}{3} \cdot \frac{s}{3}$

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 $\frac{3 r}{4} \cdot \frac{5 r}{4}$ .

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Step 3.1.1.1

Multiply $\frac{3 r}{4}$ by $\frac{5 r}{4}$ .

$\frac{3 r (5 r)}{4 \cdot 4} + \frac{3 r}{4} \cdot \frac{s}{3} - \frac{2 s}{3} \cdot \frac{5 r}{4} - \frac{2 s}{3} \cdot \frac{s}{3}$

Step 3.1.1.2

Multiply $5$ by $3$ .

$\frac{15 r \cdot r}{4 \cdot 4} + \frac{3 r}{4} \cdot \frac{s}{3} - \frac{2 s}{3} \cdot \frac{5 r}{4} - \frac{2 s}{3} \cdot \frac{s}{3}$

Step 3.1.1.3

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

$\frac{15 (r^{1} r)}{4 \cdot 4} + \frac{3 r}{4} \cdot \frac{s}{3} - \frac{2 s}{3} \cdot \frac{5 r}{4} - \frac{2 s}{3} \cdot \frac{s}{3}$

Step 3.1.1.4

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

$\frac{15 (r^{1} r^{1})}{4 \cdot 4} + \frac{3 r}{4} \cdot \frac{s}{3} - \frac{2 s}{3} \cdot \frac{5 r}{4} - \frac{2 s}{3} \cdot \frac{s}{3}$

Step 3.1.1.5

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

$\frac{15 r^{1 + 1}}{4 \cdot 4} + \frac{3 r}{4} \cdot \frac{s}{3} - \frac{2 s}{3} \cdot \frac{5 r}{4} - \frac{2 s}{3} \cdot \frac{s}{3}$

Step 3.1.1.6

Add $1$ and $1$ .

$\frac{15 r^{2}}{4 \cdot 4} + \frac{3 r}{4} \cdot \frac{s}{3} - \frac{2 s}{3} \cdot \frac{5 r}{4} - \frac{2 s}{3} \cdot \frac{s}{3}$

Step 3.1.1.7

Multiply $4$ by $4$ .

$\frac{15 r^{2}}{16} + \frac{3 r}{4} \cdot \frac{s}{3} - \frac{2 s}{3} \cdot \frac{5 r}{4} - \frac{2 s}{3} \cdot \frac{s}{3}$

Step 3.1.2

Cancel the common factor of $3$ .

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Step 3.1.2.1

Factor $3$ out of $3 r$ .

$\frac{15 r^{2}}{16} + \frac{3 (r)}{4} \cdot \frac{s}{3} - \frac{2 s}{3} \cdot \frac{5 r}{4} - \frac{2 s}{3} \cdot \frac{s}{3}$

Step 3.1.2.2

Cancel the common factor.

$\frac{15 r^{2}}{16} + \frac{3 r}{4} \cdot \frac{s}{3} - \frac{2 s}{3} \cdot \frac{5 r}{4} - \frac{2 s}{3} \cdot \frac{s}{3}$

Step 3.1.2.3

Rewrite the expression.

$\frac{15 r^{2}}{16} + \frac{r}{4} s - \frac{2 s}{3} \cdot \frac{5 r}{4} - \frac{2 s}{3} \cdot \frac{s}{3}$

Step 3.1.3

Combine $\frac{r}{4}$ and $s$ .

$\frac{15 r^{2}}{16} + \frac{r s}{4} - \frac{2 s}{3} \cdot \frac{5 r}{4} - \frac{2 s}{3} \cdot \frac{s}{3}$

Step 3.1.4

Cancel the common factor of $2$ .

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Step 3.1.4.1

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

$\frac{15 r^{2}}{16} + \frac{r s}{4} + \frac{- 2 s}{3} \cdot \frac{5 r}{4} - \frac{2 s}{3} \cdot \frac{s}{3}$

Step 3.1.4.2

Factor $2$ out of $- 2 s$ .

$\frac{15 r^{2}}{16} + \frac{r s}{4} + \frac{2 (- s)}{3} \cdot \frac{5 r}{4} - \frac{2 s}{3} \cdot \frac{s}{3}$

Step 3.1.4.3

Factor $2$ out of $4$ .

$\frac{15 r^{2}}{16} + \frac{r s}{4} + \frac{2 (- s)}{3} \cdot \frac{5 r}{2 (2)} - \frac{2 s}{3} \cdot \frac{s}{3}$

Step 3.1.4.4

Cancel the common factor.

$\frac{15 r^{2}}{16} + \frac{r s}{4} + \frac{2 (- s)}{3} \cdot \frac{5 r}{2 \cdot 2} - \frac{2 s}{3} \cdot \frac{s}{3}$

Step 3.1.4.5

Rewrite the expression.

$\frac{15 r^{2}}{16} + \frac{r s}{4} + \frac{- s}{3} \cdot \frac{5 r}{2} - \frac{2 s}{3} \cdot \frac{s}{3}$

Step 3.1.5

Multiply $\frac{- s}{3}$ by $\frac{5 r}{2}$ .

$\frac{15 r^{2}}{16} + \frac{r s}{4} + \frac{- s (5 r)}{3 \cdot 2} - \frac{2 s}{3} \cdot \frac{s}{3}$

Step 3.1.6

Multiply $5$ by $- 1$ .

$\frac{15 r^{2}}{16} + \frac{r s}{4} + \frac{- 5 s r}{3 \cdot 2} - \frac{2 s}{3} \cdot \frac{s}{3}$

Step 3.1.7

Multiply $3$ by $2$ .

$\frac{15 r^{2}}{16} + \frac{r s}{4} + \frac{- 5 s r}{6} - \frac{2 s}{3} \cdot \frac{s}{3}$

Step 3.1.8

Move the negative in front of the fraction.

$\frac{15 r^{2}}{16} + \frac{r s}{4} - \frac{5 s r}{6} - \frac{2 s}{3} \cdot \frac{s}{3}$

Step 3.1.9

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

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Step 3.1.9.1

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

$\frac{15 r^{2}}{16} + \frac{r s}{4} - \frac{5 s r}{6} - \frac{s (2 s)}{3 \cdot 3}$

Step 3.1.9.2

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

$\frac{15 r^{2}}{16} + \frac{r s}{4} - \frac{5 s r}{6} - \frac{2 (s^{1} s)}{3 \cdot 3}$

Step 3.1.9.3

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

$\frac{15 r^{2}}{16} + \frac{r s}{4} - \frac{5 s r}{6} - \frac{2 (s^{1} s^{1})}{3 \cdot 3}$

Step 3.1.9.4

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

$\frac{15 r^{2}}{16} + \frac{r s}{4} - \frac{5 s r}{6} - \frac{2 s^{1 + 1}}{3 \cdot 3}$

Step 3.1.9.5

Add $1$ and $1$ .

$\frac{15 r^{2}}{16} + \frac{r s}{4} - \frac{5 s r}{6} - \frac{2 s^{2}}{3 \cdot 3}$

Step 3.1.9.6

Multiply $3$ by $3$ .

$\frac{15 r^{2}}{16} + \frac{r s}{4} - \frac{5 s r}{6} - \frac{2 s^{2}}{9}$

Step 3.2

Subtract $\frac{5 s r}{6}$ from $\frac{r s}{4}$ .

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Step 3.2.1

Move $s$ .

$\frac{15 r^{2}}{16} + \frac{r s}{4} - \frac{5 r s}{6} - \frac{2 s^{2}}{9}$

Step 3.2.2

To write $\frac{r s}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{15 r^{2}}{16} + \frac{r s}{4} \cdot \frac{3}{3} - \frac{5 r s}{6} - \frac{2 s^{2}}{9}$

Step 3.2.3

To write $- \frac{5 r s}{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{15 r^{2}}{16} + \frac{r s}{4} \cdot \frac{3}{3} - \frac{5 r s}{6} \cdot \frac{2}{2} - \frac{2 s^{2}}{9}$

Step 3.2.4

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.2.4.1

Multiply $\frac{r s}{4}$ by $\frac{3}{3}$ .

$\frac{15 r^{2}}{16} + \frac{r s \cdot 3}{4 \cdot 3} - \frac{5 r s}{6} \cdot \frac{2}{2} - \frac{2 s^{2}}{9}$

Step 3.2.4.2

Multiply $4$ by $3$ .

$\frac{15 r^{2}}{16} + \frac{r s \cdot 3}{12} - \frac{5 r s}{6} \cdot \frac{2}{2} - \frac{2 s^{2}}{9}$

Step 3.2.4.3

Multiply $\frac{5 r s}{6}$ by $\frac{2}{2}$ .

$\frac{15 r^{2}}{16} + \frac{r s \cdot 3}{12} - \frac{5 r s \cdot 2}{6 \cdot 2} - \frac{2 s^{2}}{9}$

Step 3.2.4.4

Multiply $6$ by $2$ .

$\frac{15 r^{2}}{16} + \frac{r s \cdot 3}{12} - \frac{5 r s \cdot 2}{12} - \frac{2 s^{2}}{9}$

Step 3.2.5

Combine the numerators over the common denominator.

$\frac{15 r^{2}}{16} + \frac{r s \cdot 3 - 5 r s \cdot 2}{12} - \frac{2 s^{2}}{9}$

Step 3.3

To write $\frac{15 r^{2}}{16}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{15 r^{2}}{16} \cdot \frac{3}{3} + \frac{r s \cdot 3 - 5 r s \cdot 2}{12} - \frac{2 s^{2}}{9}$

Step 3.4

To write $\frac{r s \cdot 3 - 5 r s \cdot 2}{12}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{15 r^{2}}{16} \cdot \frac{3}{3} + \frac{r s \cdot 3 - 5 r s \cdot 2}{12} \cdot \frac{4}{4} - \frac{2 s^{2}}{9}$

Step 3.5

Write each expression with a common denominator of $48$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.5.1

Multiply $\frac{15 r^{2}}{16}$ by $\frac{3}{3}$ .

$\frac{15 r^{2} \cdot 3}{16 \cdot 3} + \frac{r s \cdot 3 - 5 r s \cdot 2}{12} \cdot \frac{4}{4} - \frac{2 s^{2}}{9}$

Step 3.5.2

Multiply $16$ by $3$ .

$\frac{15 r^{2} \cdot 3}{48} + \frac{r s \cdot 3 - 5 r s \cdot 2}{12} \cdot \frac{4}{4} - \frac{2 s^{2}}{9}$

Step 3.5.3

Multiply $\frac{r s \cdot 3 - 5 r s \cdot 2}{12}$ by $\frac{4}{4}$ .

$\frac{15 r^{2} \cdot 3}{48} + \frac{(r s \cdot 3 - 5 r s \cdot 2) \cdot 4}{12 \cdot 4} - \frac{2 s^{2}}{9}$

Step 3.5.4

Multiply $12$ by $4$ .

$\frac{15 r^{2} \cdot 3}{48} + \frac{(r s \cdot 3 - 5 r s \cdot 2) \cdot 4}{48} - \frac{2 s^{2}}{9}$

Step 3.6

Combine the numerators over the common denominator.

$\frac{15 r^{2} \cdot 3 + (r s \cdot 3 - 5 r s \cdot 2) \cdot 4}{48} - \frac{2 s^{2}}{9}$

Step 3.7

Reorder terms.

$\frac{3 \cdot 15 r^{2} + 4 (r s \cdot 3 - 5 r s \cdot 2)}{48} - \frac{2 s^{2}}{9}$

Step 3.8

Combine $\frac{3 \cdot 15 r^{2} + 4 (r s \cdot 3 - 5 r s \cdot 2)}{48}$ and $- \frac{2 s^{2}}{9}$ using a common denominator.

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Step 3.8.1

Reorder $3 \cdot 15 r^{2}$ and $4 (3 \cdot r s - 5 \cdot 2 r s)$ .

$\frac{4 (3 \cdot r s - 5 \cdot 2 r s) + 3 \cdot 15 r^{2}}{48} - \frac{2 s^{2}}{9}$

Step 3.8.2

To write $\frac{4 (3 \cdot r s - 5 \cdot 2 r s) + 3 \cdot 15 r^{2}}{48}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{4 (3 \cdot r s - 5 \cdot 2 r s) + 3 \cdot 15 r^{2}}{48} \cdot \frac{3}{3} - \frac{2 s^{2}}{9}$

Step 3.8.3

To write $- \frac{2 s^{2}}{9}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\frac{4 (3 \cdot r s - 5 \cdot 2 r s) + 3 \cdot 15 r^{2}}{48} \cdot \frac{3}{3} - \frac{2 s^{2}}{9} \cdot \frac{16}{16}$

Step 3.8.4

Write each expression with a common denominator of $144$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.8.4.1

Multiply $\frac{4 (3 \cdot r s - 5 \cdot 2 r s) + 3 \cdot 15 r^{2}}{48}$ by $\frac{3}{3}$ .

$\frac{(4 (3 \cdot r s - 5 \cdot 2 r s) + 3 \cdot 15 r^{2}) \cdot 3}{48 \cdot 3} - \frac{2 s^{2}}{9} \cdot \frac{16}{16}$

Step 3.8.4.2

Multiply $48$ by $3$ .

$\frac{(4 (3 \cdot r s - 5 \cdot 2 r s) + 3 \cdot 15 r^{2}) \cdot 3}{144} - \frac{2 s^{2}}{9} \cdot \frac{16}{16}$

Step 3.8.4.3

Multiply $\frac{2 s^{2}}{9}$ by $\frac{16}{16}$ .

$\frac{(4 (3 \cdot r s - 5 \cdot 2 r s) + 3 \cdot 15 r^{2}) \cdot 3}{144} - \frac{2 s^{2} \cdot 16}{9 \cdot 16}$

Step 3.8.4.4

Multiply $9$ by $16$ .

$\frac{(4 (3 \cdot r s - 5 \cdot 2 r s) + 3 \cdot 15 r^{2}) \cdot 3}{144} - \frac{2 s^{2} \cdot 16}{144}$

Step 3.8.5

Combine the numerators over the common denominator.

$\frac{(4 (3 \cdot r s - 5 \cdot 2 r s) + 3 \cdot 15 r^{2}) \cdot 3 - 2 s^{2} \cdot 16}{144}$

Step 4

Simplify the numerator.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $- 5$ by $2$ .

$\frac{(4 (3 r s - 10 r s) + 3 \cdot 15 r^{2}) \cdot 3 - 2 s^{2} \cdot 16}{144}$

Step 4.1.2

Subtract $10 r s$ from $3 r s$ .

$\frac{(4 (- 7 r s) + 3 \cdot 15 r^{2}) \cdot 3 - 2 s^{2} \cdot 16}{144}$

Step 4.1.3

Multiply $- 7$ by $4$ .

$\frac{(- 28 (r s) + 3 \cdot 15 r^{2}) \cdot 3 - 2 s^{2} \cdot 16}{144}$

Step 4.1.4

Multiply $3$ by $15$ .

$\frac{(- 28 r s + 45 r^{2}) \cdot 3 - 2 s^{2} \cdot 16}{144}$

Step 4.2

Apply the distributive property.

$\frac{- 28 r s \cdot 3 + 45 r^{2} \cdot 3 - 2 s^{2} \cdot 16}{144}$

Step 4.3

Multiply $3$ by $- 28$ .

$\frac{- 84 r s + 45 r^{2} \cdot 3 - 2 s^{2} \cdot 16}{144}$

Step 4.4

Multiply $3$ by $45$ .

$\frac{- 84 r s + 135 r^{2} - 2 s^{2} \cdot 16}{144}$

Step 4.5

Multiply $16$ by $- 2$ .

$\frac{- 84 r s + 135 r^{2} - 32 s^{2}}{144}$

Step 4.6

Factor by grouping.

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Step 4.6.1

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 $a \cdot c = 135 \cdot - 32 = - 4320$ and whose sum is $b = - 84$ .

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Step 4.6.1.1

Reorder terms.

$\frac{135 r^{2} - 32 s^{2} - 84 r s}{144}$

Step 4.6.1.2

Reorder $- 32 s^{2}$ and $- 84 r s$ .

$\frac{135 r^{2} - 84 r s - 32 s^{2}}{144}$

Step 4.6.1.3

Factor $- 84$ out of $- 84 r s$ .

$\frac{135 r^{2} - 84 (r s) - 32 s^{2}}{144}$

Step 4.6.1.4

Rewrite $- 84$ as $36$ plus $- 120$

$\frac{135 r^{2} + (36 - 120) (r s) - 32 s^{2}}{144}$

Step 4.6.1.5

Apply the distributive property.

$\frac{135 r^{2} + 36 (r s) - 120 (r s) - 32 s^{2}}{144}$

Step 4.6.1.6

Move parentheses.

$\frac{135 r^{2} + 36 r s - 120 r s - 32 s^{2}}{144}$

Step 4.6.2

Factor out the greatest common factor from each group.

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Step 4.6.2.1

Group the first two terms and the last two terms.

$\frac{(135 r^{2} + 36 r s) - 120 r s - 32 s^{2}}{144}$

Step 4.6.2.2

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

$\frac{9 r (15 r + 4 s) - 8 s (15 r + 4 s)}{144}$

Step 4.6.3

Factor the polynomial by factoring out the greatest common factor, $15 r + 4 s$ .

$\frac{(15 r + 4 s) (9 r - 8 s)}{144}$