Basic Math Examples

$9 \cdot ({(1 - c)}^{2}) - \frac{45}{4} \cdot (\frac{1}{2} \cdot (c^{2} + 1) + \frac{3}{19} \cdot {(1 - c)}^{2})$

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(1 - c)}^{2}$ as $(1 - c) (1 - c)$ .

$9 \cdot ((1 - c) (1 - c)) - \frac{45}{4} \cdot (\frac{1}{2} \cdot (c^{2} + 1) + \frac{3}{19} \cdot {(1 - c)}^{2})$

Step 1.2

Expand $(1 - c) (1 - c)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$9 \cdot (1 (1 - c) - c (1 - c)) - \frac{45}{4} \cdot (\frac{1}{2} \cdot (c^{2} + 1) + \frac{3}{19} \cdot {(1 - c)}^{2})$

Step 1.2.2

Apply the distributive property.

$9 \cdot (1 \cdot 1 + 1 (- c) - c (1 - c)) - \frac{45}{4} \cdot (\frac{1}{2} \cdot (c^{2} + 1) + \frac{3}{19} \cdot {(1 - c)}^{2})$

Step 1.2.3

Apply the distributive property.

$9 \cdot (1 \cdot 1 + 1 (- c) - c \cdot 1 - c (- c)) - \frac{45}{4} \cdot (\frac{1}{2} \cdot (c^{2} + 1) + \frac{3}{19} \cdot {(1 - c)}^{2})$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $1$ by $1$ .

$9 \cdot (1 + 1 (- c) - c \cdot 1 - c (- c)) - \frac{45}{4} \cdot (\frac{1}{2} \cdot (c^{2} + 1) + \frac{3}{19} \cdot {(1 - c)}^{2})$

Step 1.3.1.2

Multiply $- c$ by $1$ .

$9 \cdot (1 - c - c \cdot 1 - c (- c)) - \frac{45}{4} \cdot (\frac{1}{2} \cdot (c^{2} + 1) + \frac{3}{19} \cdot {(1 - c)}^{2})$

Step 1.3.1.3

Multiply $- 1$ by $1$ .

$9 \cdot (1 - c - c - c (- c)) - \frac{45}{4} \cdot (\frac{1}{2} \cdot (c^{2} + 1) + \frac{3}{19} \cdot {(1 - c)}^{2})$

Step 1.3.1.4

Rewrite using the commutative property of multiplication.

$9 \cdot (1 - c - c - 1 \cdot - 1 c \cdot c) - \frac{45}{4} \cdot (\frac{1}{2} \cdot (c^{2} + 1) + \frac{3}{19} \cdot {(1 - c)}^{2})$

Step 1.3.1.5

Multiply $c$ by $c$ by adding the exponents.

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Step 1.3.1.5.1

Move $c$ .

$9 \cdot (1 - c - c - 1 \cdot - 1 (c \cdot c)) - \frac{45}{4} \cdot (\frac{1}{2} \cdot (c^{2} + 1) + \frac{3}{19} \cdot {(1 - c)}^{2})$

Step 1.3.1.5.2

Multiply $c$ by $c$ .

$9 \cdot (1 - c - c - 1 \cdot - 1 c^{2}) - \frac{45}{4} \cdot (\frac{1}{2} \cdot (c^{2} + 1) + \frac{3}{19} \cdot {(1 - c)}^{2})$

Step 1.3.1.6

Multiply $- 1$ by $- 1$ .

$9 \cdot (1 - c - c + 1 c^{2}) - \frac{45}{4} \cdot (\frac{1}{2} \cdot (c^{2} + 1) + \frac{3}{19} \cdot {(1 - c)}^{2})$

Step 1.3.1.7

Multiply $c^{2}$ by $1$ .

$9 \cdot (1 - c - c + c^{2}) - \frac{45}{4} \cdot (\frac{1}{2} \cdot (c^{2} + 1) + \frac{3}{19} \cdot {(1 - c)}^{2})$

Step 1.3.2

Subtract $c$ from $- c$ .

$9 \cdot (1 - 2 c + c^{2}) - \frac{45}{4} \cdot (\frac{1}{2} \cdot (c^{2} + 1) + \frac{3}{19} \cdot {(1 - c)}^{2})$

Step 1.4

Apply the distributive property.

$9 \cdot 1 + 9 (- 2 c) + 9 c^{2} - \frac{45}{4} \cdot (\frac{1}{2} \cdot (c^{2} + 1) + \frac{3}{19} \cdot {(1 - c)}^{2})$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $9$ by $1$ .

$9 + 9 (- 2 c) + 9 c^{2} - \frac{45}{4} \cdot (\frac{1}{2} \cdot (c^{2} + 1) + \frac{3}{19} \cdot {(1 - c)}^{2})$

Step 1.5.2

Multiply $- 2$ by $9$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{1}{2} \cdot (c^{2} + 1) + \frac{3}{19} \cdot {(1 - c)}^{2})$

Step 1.6

Simplify each term.

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Step 1.6.1

Apply the distributive property.

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{1}{2} c^{2} + \frac{1}{2} \cdot 1 + \frac{3}{19} \cdot {(1 - c)}^{2})$

Step 1.6.2

Combine $\frac{1}{2}$ and $c^{2}$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{c^{2}}{2} + \frac{1}{2} \cdot 1 + \frac{3}{19} \cdot {(1 - c)}^{2})$

Step 1.6.3

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

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{c^{2}}{2} + \frac{1}{2} + \frac{3}{19} \cdot {(1 - c)}^{2})$

Step 1.6.4

Rewrite ${(1 - c)}^{2}$ as $(1 - c) (1 - c)$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{c^{2}}{2} + \frac{1}{2} + \frac{3}{19} \cdot ((1 - c) (1 - c)))$

Step 1.6.5

Expand $(1 - c) (1 - c)$ using the FOIL Method.

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Step 1.6.5.1

Apply the distributive property.

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{c^{2}}{2} + \frac{1}{2} + \frac{3}{19} \cdot (1 (1 - c) - c (1 - c)))$

Step 1.6.5.2

Apply the distributive property.

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{c^{2}}{2} + \frac{1}{2} + \frac{3}{19} \cdot (1 \cdot 1 + 1 (- c) - c (1 - c)))$

Step 1.6.5.3

Apply the distributive property.

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{c^{2}}{2} + \frac{1}{2} + \frac{3}{19} \cdot (1 \cdot 1 + 1 (- c) - c \cdot 1 - c (- c)))$

Step 1.6.6

Simplify and combine like terms.

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Step 1.6.6.1

Simplify each term.

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Step 1.6.6.1.1

Multiply $1$ by $1$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{c^{2}}{2} + \frac{1}{2} + \frac{3}{19} \cdot (1 + 1 (- c) - c \cdot 1 - c (- c)))$

Step 1.6.6.1.2

Multiply $- c$ by $1$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{c^{2}}{2} + \frac{1}{2} + \frac{3}{19} \cdot (1 - c - c \cdot 1 - c (- c)))$

Step 1.6.6.1.3

Multiply $- 1$ by $1$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{c^{2}}{2} + \frac{1}{2} + \frac{3}{19} \cdot (1 - c - c - c (- c)))$

Step 1.6.6.1.4

Rewrite using the commutative property of multiplication.

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{c^{2}}{2} + \frac{1}{2} + \frac{3}{19} \cdot (1 - c - c - 1 \cdot - 1 c \cdot c))$

Step 1.6.6.1.5

Multiply $c$ by $c$ by adding the exponents.

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Step 1.6.6.1.5.1

Move $c$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{c^{2}}{2} + \frac{1}{2} + \frac{3}{19} \cdot (1 - c - c - 1 \cdot - 1 (c \cdot c)))$

Step 1.6.6.1.5.2

Multiply $c$ by $c$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{c^{2}}{2} + \frac{1}{2} + \frac{3}{19} \cdot (1 - c - c - 1 \cdot - 1 c^{2}))$

Step 1.6.6.1.6

Multiply $- 1$ by $- 1$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{c^{2}}{2} + \frac{1}{2} + \frac{3}{19} \cdot (1 - c - c + 1 c^{2}))$

Step 1.6.6.1.7

Multiply $c^{2}$ by $1$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{c^{2}}{2} + \frac{1}{2} + \frac{3}{19} \cdot (1 - c - c + c^{2}))$

Step 1.6.6.2

Subtract $c$ from $- c$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{c^{2}}{2} + \frac{1}{2} + \frac{3}{19} \cdot (1 - 2 c + c^{2}))$

Step 1.6.7

Apply the distributive property.

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{c^{2}}{2} + \frac{1}{2} + \frac{3}{19} \cdot 1 + \frac{3}{19} (- 2 c) + \frac{3}{19} c^{2})$

Step 1.6.8

Simplify.

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Step 1.6.8.1

Multiply $\frac{3}{19}$ by $1$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{c^{2}}{2} + \frac{1}{2} + \frac{3}{19} + \frac{3}{19} (- 2 c) + \frac{3}{19} c^{2})$

Step 1.6.8.2

Multiply $\frac{3}{19} (- 2 c)$ .

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Step 1.6.8.2.1

Combine $- 2$ and $\frac{3}{19}$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{c^{2}}{2} + \frac{1}{2} + \frac{3}{19} + \frac{- 2 \cdot 3}{19} c + \frac{3}{19} c^{2})$

Step 1.6.8.2.2

Multiply $- 2$ by $3$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{c^{2}}{2} + \frac{1}{2} + \frac{3}{19} + \frac{- 6}{19} c + \frac{3}{19} c^{2})$

Step 1.6.8.2.3

Combine $\frac{- 6}{19}$ and $c$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{c^{2}}{2} + \frac{1}{2} + \frac{3}{19} + \frac{- 6 c}{19} + \frac{3}{19} c^{2})$

Step 1.6.8.3

Combine $\frac{3}{19}$ and $c^{2}$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{c^{2}}{2} + \frac{1}{2} + \frac{3}{19} + \frac{- 6 c}{19} + \frac{3 c^{2}}{19})$

Step 1.6.9

Move the negative in front of the fraction.

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{c^{2}}{2} + \frac{1}{2} + \frac{3}{19} - \frac{6 c}{19} + \frac{3 c^{2}}{19})$

Step 1.7

To write $\frac{c^{2}}{2}$ as a fraction with a common denominator, multiply by $\frac{19}{19}$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{1}{2} + \frac{3}{19} - \frac{6 c}{19} + \frac{c^{2}}{2} \cdot \frac{19}{19} + \frac{3 c^{2}}{19})$

Step 1.8

To write $\frac{3 c^{2}}{19}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{1}{2} + \frac{3}{19} - \frac{6 c}{19} + \frac{c^{2}}{2} \cdot \frac{19}{19} + \frac{3 c^{2}}{19} \cdot \frac{2}{2})$

Step 1.9

Write each expression with a common denominator of $38$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.9.1

Multiply $\frac{c^{2}}{2}$ by $\frac{19}{19}$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{1}{2} + \frac{3}{19} - \frac{6 c}{19} + \frac{c^{2} \cdot 19}{2 \cdot 19} + \frac{3 c^{2}}{19} \cdot \frac{2}{2})$

Step 1.9.2

Multiply $2$ by $19$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{1}{2} + \frac{3}{19} - \frac{6 c}{19} + \frac{c^{2} \cdot 19}{38} + \frac{3 c^{2}}{19} \cdot \frac{2}{2})$

Step 1.9.3

Multiply $\frac{3 c^{2}}{19}$ by $\frac{2}{2}$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{1}{2} + \frac{3}{19} - \frac{6 c}{19} + \frac{c^{2} \cdot 19}{38} + \frac{3 c^{2} \cdot 2}{19 \cdot 2})$

Step 1.9.4

Multiply $19$ by $2$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{1}{2} + \frac{3}{19} - \frac{6 c}{19} + \frac{c^{2} \cdot 19}{38} + \frac{3 c^{2} \cdot 2}{38})$

Step 1.10

Combine the numerators over the common denominator.

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{1}{2} + \frac{3}{19} - \frac{6 c}{19} + \frac{c^{2} \cdot 19 + 3 c^{2} \cdot 2}{38})$

Step 1.11

Simplify each term.

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Step 1.11.1

Simplify the numerator.

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Step 1.11.1.1

Factor $c^{2}$ out of $c^{2} \cdot 19 + 3 c^{2} \cdot 2$ .

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Step 1.11.1.1.1

Factor $c^{2}$ out of $c^{2} \cdot 19$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{1}{2} + \frac{3}{19} - \frac{6 c}{19} + \frac{c^{2} (19) + 3 c^{2} \cdot 2}{38})$

Step 1.11.1.1.2

Factor $c^{2}$ out of $3 c^{2} \cdot 2$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{1}{2} + \frac{3}{19} - \frac{6 c}{19} + \frac{c^{2} (19) + c^{2} (3 \cdot 2)}{38})$

Step 1.11.1.1.3

Factor $c^{2}$ out of $c^{2} (19) + c^{2} (3 \cdot 2)$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{1}{2} + \frac{3}{19} - \frac{6 c}{19} + \frac{c^{2} (19 + 3 \cdot 2)}{38})$

Step 1.11.1.2

Multiply $3$ by $2$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{1}{2} + \frac{3}{19} - \frac{6 c}{19} + \frac{c^{2} (19 + 6)}{38})$

Step 1.11.1.3

Add $19$ and $6$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{1}{2} + \frac{3}{19} - \frac{6 c}{19} + \frac{c^{2} \cdot 25}{38})$

Step 1.11.2

Move $25$ to the left of $c^{2}$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{1}{2} + \frac{3}{19} - \frac{6 c}{19} + \frac{25 c^{2}}{38})$

Step 1.12

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{19}{19}$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{1}{2} \cdot \frac{19}{19} + \frac{3}{19} - \frac{6 c}{19} + \frac{25 c^{2}}{38})$

Step 1.13

To write $\frac{3}{19}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{1}{2} \cdot \frac{19}{19} + \frac{3}{19} \cdot \frac{2}{2} - \frac{6 c}{19} + \frac{25 c^{2}}{38})$

Step 1.14

Write each expression with a common denominator of $38$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.14.1

Multiply $\frac{1}{2}$ by $\frac{19}{19}$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{19}{2 \cdot 19} + \frac{3}{19} \cdot \frac{2}{2} - \frac{6 c}{19} + \frac{25 c^{2}}{38})$

Step 1.14.2

Multiply $2$ by $19$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{19}{38} + \frac{3}{19} \cdot \frac{2}{2} - \frac{6 c}{19} + \frac{25 c^{2}}{38})$

Step 1.14.3

Multiply $\frac{3}{19}$ by $\frac{2}{2}$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{19}{38} + \frac{3 \cdot 2}{19 \cdot 2} - \frac{6 c}{19} + \frac{25 c^{2}}{38})$

Step 1.14.4

Multiply $19$ by $2$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{19}{38} + \frac{3 \cdot 2}{38} - \frac{6 c}{19} + \frac{25 c^{2}}{38})$

Step 1.15

Combine the numerators over the common denominator.

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{19 + 3 \cdot 2}{38} - \frac{6 c}{19} + \frac{25 c^{2}}{38})$

Step 1.16

Simplify the numerator.

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Step 1.16.1

Multiply $3$ by $2$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{19 + 6}{38} - \frac{6 c}{19} + \frac{25 c^{2}}{38})$

Step 1.16.2

Add $19$ and $6$ .

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot (\frac{25}{38} - \frac{6 c}{19} + \frac{25 c^{2}}{38})$

Step 1.17

Apply the distributive property.

$9 - 18 c + 9 c^{2} - \frac{45}{4} \cdot \frac{25}{38} - \frac{45}{4} (- \frac{6 c}{19}) - \frac{45}{4} \cdot \frac{25 c^{2}}{38}$

Step 1.18

Simplify.

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Step 1.18.1

Multiply $- \frac{45}{4} \cdot \frac{25}{38}$ .

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Step 1.18.1.1

Multiply $\frac{25}{38}$ by $\frac{45}{4}$ .

$9 - 18 c + 9 c^{2} - \frac{25 \cdot 45}{38 \cdot 4} - \frac{45}{4} (- \frac{6 c}{19}) - \frac{45}{4} \cdot \frac{25 c^{2}}{38}$

Step 1.18.1.2

Multiply $25$ by $45$ .

$9 - 18 c + 9 c^{2} - \frac{1125}{38 \cdot 4} - \frac{45}{4} (- \frac{6 c}{19}) - \frac{45}{4} \cdot \frac{25 c^{2}}{38}$

Step 1.18.1.3

Multiply $38$ by $4$ .

$9 - 18 c + 9 c^{2} - \frac{1125}{152} - \frac{45}{4} (- \frac{6 c}{19}) - \frac{45}{4} \cdot \frac{25 c^{2}}{38}$

Step 1.18.2

Cancel the common factor of $2$ .

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Step 1.18.2.1

Move the leading negative in $- \frac{45}{4}$ into the numerator.

$9 - 18 c + 9 c^{2} - \frac{1125}{152} + \frac{- 45}{4} (- \frac{6 c}{19}) - \frac{45}{4} \cdot \frac{25 c^{2}}{38}$

Step 1.18.2.2

Move the leading negative in $- \frac{6 c}{19}$ into the numerator.

$9 - 18 c + 9 c^{2} - \frac{1125}{152} + \frac{- 45}{4} \cdot \frac{- 6 c}{19} - \frac{45}{4} \cdot \frac{25 c^{2}}{38}$

Step 1.18.2.3

Factor $2$ out of $4$ .

$9 - 18 c + 9 c^{2} - \frac{1125}{152} + \frac{- 45}{2 (2)} \cdot \frac{- 6 c}{19} - \frac{45}{4} \cdot \frac{25 c^{2}}{38}$

Step 1.18.2.4

Factor $2$ out of $- 6 c$ .

$9 - 18 c + 9 c^{2} - \frac{1125}{152} + \frac{- 45}{2 (2)} \cdot \frac{2 (- 3 c)}{19} - \frac{45}{4} \cdot \frac{25 c^{2}}{38}$

Step 1.18.2.5

Cancel the common factor.

$9 - 18 c + 9 c^{2} - \frac{1125}{152} + \frac{- 45}{2 \cdot 2} \cdot \frac{2 (- 3 c)}{19} - \frac{45}{4} \cdot \frac{25 c^{2}}{38}$

Step 1.18.2.6

Rewrite the expression.

$9 - 18 c + 9 c^{2} - \frac{1125}{152} + \frac{- 45}{2} \cdot \frac{- 3 c}{19} - \frac{45}{4} \cdot \frac{25 c^{2}}{38}$

Step 1.18.3

Multiply $\frac{- 45}{2}$ by $\frac{- 3 c}{19}$ .

$9 - 18 c + 9 c^{2} - \frac{1125}{152} + \frac{- 45 (- 3 c)}{2 \cdot 19} - \frac{45}{4} \cdot \frac{25 c^{2}}{38}$

Step 1.18.4

Multiply $- 3$ by $- 45$ .

$9 - 18 c + 9 c^{2} - \frac{1125}{152} + \frac{135 c}{2 \cdot 19} - \frac{45}{4} \cdot \frac{25 c^{2}}{38}$

Step 1.18.5

Multiply $2$ by $19$ .

$9 - 18 c + 9 c^{2} - \frac{1125}{152} + \frac{135 c}{38} - \frac{45}{4} \cdot \frac{25 c^{2}}{38}$

Step 1.18.6

Multiply $- \frac{45}{4} \cdot \frac{25 c^{2}}{38}$ .

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Step 1.18.6.1

Multiply $\frac{25 c^{2}}{38}$ by $\frac{45}{4}$ .

$9 - 18 c + 9 c^{2} - \frac{1125}{152} + \frac{135 c}{38} - \frac{25 c^{2} \cdot 45}{38 \cdot 4}$

Step 1.18.6.2

Multiply $45$ by $25$ .

$9 - 18 c + 9 c^{2} - \frac{1125}{152} + \frac{135 c}{38} - \frac{1125 c^{2}}{38 \cdot 4}$

Step 1.18.6.3

Multiply $38$ by $4$ .

$9 - 18 c + 9 c^{2} - \frac{1125}{152} + \frac{135 c}{38} - \frac{1125 c^{2}}{152}$

Step 2

To write $9$ as a fraction with a common denominator, multiply by $\frac{152}{152}$ .

$- 18 c + 9 c^{2} + 9 \cdot \frac{152}{152} - \frac{1125}{152} + \frac{135 c}{38} - \frac{1125 c^{2}}{152}$

Step 3

Combine $9$ and $\frac{152}{152}$ .

$- 18 c + 9 c^{2} + \frac{9 \cdot 152}{152} - \frac{1125}{152} + \frac{135 c}{38} - \frac{1125 c^{2}}{152}$

Step 4

Combine the numerators over the common denominator.

$- 18 c + 9 c^{2} + \frac{9 \cdot 152 - 1125}{152} + \frac{135 c}{38} - \frac{1125 c^{2}}{152}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $9$ by $152$ .

$- 18 c + 9 c^{2} + \frac{1368 - 1125}{152} + \frac{135 c}{38} - \frac{1125 c^{2}}{152}$

Step 5.2

Subtract $1125$ from $1368$ .

$- 18 c + 9 c^{2} + \frac{243}{152} + \frac{135 c}{38} - \frac{1125 c^{2}}{152}$

Step 6

To write $- 18 c$ as a fraction with a common denominator, multiply by $\frac{38}{38}$ .

$9 c^{2} + \frac{243}{152} - 18 c \cdot \frac{38}{38} + \frac{135 c}{38} - \frac{1125 c^{2}}{152}$

Step 7

Simplify terms.

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Step 7.1

Combine $- 18 c$ and $\frac{38}{38}$ .

$9 c^{2} + \frac{243}{152} + \frac{- 18 c \cdot 38}{38} + \frac{135 c}{38} - \frac{1125 c^{2}}{152}$

Step 7.2

Combine the numerators over the common denominator.

$9 c^{2} + \frac{243}{152} + \frac{- 18 c \cdot 38 + 135 c}{38} - \frac{1125 c^{2}}{152}$

Step 8

Simplify each term.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Factor $9 c$ out of $- 18 c \cdot 38 + 135 c$ .

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Step 8.1.1.1

Factor $9 c$ out of $- 18 c \cdot 38$ .

$9 c^{2} + \frac{243}{152} + \frac{9 c (- 2 \cdot 38) + 135 c}{38} - \frac{1125 c^{2}}{152}$

Step 8.1.1.2

Factor $9 c$ out of $135 c$ .

$9 c^{2} + \frac{243}{152} + \frac{9 c (- 2 \cdot 38) + 9 c (15)}{38} - \frac{1125 c^{2}}{152}$

Step 8.1.1.3

Factor $9 c$ out of $9 c (- 2 \cdot 38) + 9 c (15)$ .

$9 c^{2} + \frac{243}{152} + \frac{9 c (- 2 \cdot 38 + 15)}{38} - \frac{1125 c^{2}}{152}$

Step 8.1.2

Multiply $- 2$ by $38$ .

$9 c^{2} + \frac{243}{152} + \frac{9 c (- 76 + 15)}{38} - \frac{1125 c^{2}}{152}$

Step 8.1.3

Add $- 76$ and $15$ .

$9 c^{2} + \frac{243}{152} + \frac{9 c \cdot - 61}{38} - \frac{1125 c^{2}}{152}$

Step 8.1.4

Multiply $- 61$ by $9$ .

$9 c^{2} + \frac{243}{152} + \frac{- 549 c}{38} - \frac{1125 c^{2}}{152}$

Step 8.2

Move the negative in front of the fraction.

$9 c^{2} + \frac{243}{152} - \frac{549 c}{38} - \frac{1125 c^{2}}{152}$

Step 9

To write $9 c^{2}$ as a fraction with a common denominator, multiply by $\frac{152}{152}$ .

$\frac{243}{152} - \frac{549 c}{38} + 9 c^{2} \cdot \frac{152}{152} - \frac{1125 c^{2}}{152}$

Step 10

Simplify terms.

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Step 10.1

Combine $9 c^{2}$ and $\frac{152}{152}$ .

$\frac{243}{152} - \frac{549 c}{38} + \frac{9 c^{2} \cdot 152}{152} - \frac{1125 c^{2}}{152}$

Step 10.2

Combine the numerators over the common denominator.

$\frac{243}{152} - \frac{549 c}{38} + \frac{9 c^{2} \cdot 152 - 1125 c^{2}}{152}$

Step 10.3

Combine the numerators over the common denominator.

$\frac{243 + 9 c^{2} \cdot 152 - 1125 c^{2}}{152} + \frac{- 549 c}{38}$

Step 10.4

Multiply $152$ by $9$ .

$\frac{243 + 1368 c^{2} - 1125 c^{2}}{152} + \frac{- 549 c}{38}$

Step 10.5

Subtract $1125 c^{2}$ from $1368 c^{2}$ .

$\frac{243 + 243 c^{2}}{152} + \frac{- 549 c}{38}$

Step 11

Simplify each term.

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Step 11.1

Factor $243$ out of $243 + 243 c^{2}$ .

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Step 11.1.1

Factor $243$ out of $243$ .

$\frac{243 \cdot 1 + 243 c^{2}}{152} + \frac{- 549 c}{38}$

Step 11.1.2

Factor $243$ out of $243 \cdot 1 + 243 c^{2}$ .

$\frac{243 (1 + c^{2})}{152} + \frac{- 549 c}{38}$

Step 11.2

Move the negative in front of the fraction.

$\frac{243 (1 + c^{2})}{152} - \frac{549 c}{38}$

Step 12

To write $- \frac{549 c}{38}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{243 (1 + c^{2})}{152} - \frac{549 c}{38} \cdot \frac{4}{4}$

Step 13

Write each expression with a common denominator of $152$ , by multiplying each by an appropriate factor of $1$ .

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Step 13.1

Multiply $\frac{549 c}{38}$ by $\frac{4}{4}$ .

$\frac{243 (1 + c^{2})}{152} - \frac{549 c \cdot 4}{38 \cdot 4}$

Step 13.2

Multiply $38$ by $4$ .

$\frac{243 (1 + c^{2})}{152} - \frac{549 c \cdot 4}{152}$

Step 14

Combine the numerators over the common denominator.

$\frac{243 (1 + c^{2}) - 549 c \cdot 4}{152}$

Step 15

Simplify the numerator.

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Step 15.1

Factor $9$ out of $243 (1 + c^{2}) - 549 c \cdot 4$ .

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Step 15.1.1

Factor $9$ out of $243 (1 + c^{2})$ .

$\frac{9 (27 (1 + c^{2})) - 549 c \cdot 4}{152}$

Step 15.1.2

Factor $9$ out of $- 549 c \cdot 4$ .

$\frac{9 (27 (1 + c^{2})) + 9 (- 61 c \cdot 4)}{152}$

Step 15.1.3

Factor $9$ out of $9 (27 (1 + c^{2})) + 9 (- 61 c \cdot 4)$ .

$\frac{9 (27 (1 + c^{2}) - 61 c \cdot 4)}{152}$

Step 15.2

Apply the distributive property.

$\frac{9 (27 \cdot 1 + 27 c^{2} - 61 c \cdot 4)}{152}$

Step 15.3

Multiply $27$ by $1$ .

$\frac{9 (27 + 27 c^{2} - 61 c \cdot 4)}{152}$

Step 15.4

Multiply $4$ by $- 61$ .

$\frac{9 (27 + 27 c^{2} - 244 c)}{152}$

Step 15.5

Reorder terms.

$\frac{9 (27 c^{2} - 244 c + 27)}{152}$