Pre-Algebra Examples

$\frac{\frac{x^{2} - 14 x + 45}{x^{2} - 5 x - 36}}{\frac{x^{2} \cdot (3 x) - 10}{x^{2} - 7 x - 18}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{2} - 14 x + 45}{x^{2} - 5 x - 36} \cdot \frac{x^{2} - 7 x - 18}{x^{2} \cdot (3 x) - 10}$

Step 2

Factor $x^{2} - 14 x + 45$ using the AC method.

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Step 2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $45$ and whose sum is $- 14$ .

$- 9, - 5$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 9) (x - 5)}{x^{2} - 5 x - 36} \cdot \frac{x^{2} - 7 x - 18}{x^{2} \cdot (3 x) - 10}$

Step 3

Factor $x^{2} - 5 x - 36$ using the AC method.

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Step 3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 36$ and whose sum is $- 5$ .

$- 9, 4$

Step 3.2

Write the factored form using these integers.

$\frac{(x - 9) (x - 5)}{(x - 9) (x + 4)} \cdot \frac{x^{2} - 7 x - 18}{x^{2} \cdot (3 x) - 10}$

Step 4

Cancel the common factor of $x - 9$ .

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Step 4.1

Cancel the common factor.

$\frac{(x - 9) (x - 5)}{(x - 9) (x + 4)} \cdot \frac{x^{2} - 7 x - 18}{x^{2} \cdot (3 x) - 10}$

Step 4.2

Rewrite the expression.

$\frac{x - 5}{x + 4} \cdot \frac{x^{2} - 7 x - 18}{x^{2} \cdot (3 x) - 10}$

Step 5

Factor $x^{2} - 7 x - 18$ using the AC method.

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Step 5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 18$ and whose sum is $- 7$ .

$- 9, 2$

Step 5.2

Write the factored form using these integers.

$\frac{x - 5}{x + 4} \cdot \frac{(x - 9) (x + 2)}{x^{2} \cdot (3 x) - 10}$

Step 6

Simplify the denominator.

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Step 6.1

Rewrite using the commutative property of multiplication.

$\frac{x - 5}{x + 4} \cdot \frac{(x - 9) (x + 2)}{3 x^{2} x - 10}$

Step 6.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 6.2.1

Move $x$ .

$\frac{x - 5}{x + 4} \cdot \frac{(x - 9) (x + 2)}{3 (x \cdot x^{2}) - 10}$

Step 6.2.2

Multiply $x$ by $x^{2}$ .

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Step 6.2.2.1

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

$\frac{x - 5}{x + 4} \cdot \frac{(x - 9) (x + 2)}{3 (x^{1} x^{2}) - 10}$

Step 6.2.2.2

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

$\frac{x - 5}{x + 4} \cdot \frac{(x - 9) (x + 2)}{3 x^{1 + 2} - 10}$

Step 6.2.3

Add $1$ and $2$ .

$\frac{x - 5}{x + 4} \cdot \frac{(x - 9) (x + 2)}{3 x^{3} - 10}$

Step 7

Multiply $\frac{x - 5}{x + 4}$ by $\frac{(x - 9) (x + 2)}{3 x^{3} - 10}$ .

$\frac{(x - 5) (x - 9) (x + 2)}{(x + 4) (3 x^{3} - 10)}$

Step 8

Expand $(x - 5) (x - 9)$ using the FOIL Method.

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Step 8.1

Apply the distributive property.

$\frac{(x (x - 9) - 5 (x - 9)) (x + 2)}{(x + 4) (3 x^{3} - 10)}$

Step 8.2

Apply the distributive property.

$\frac{(x \cdot x + x \cdot - 9 - 5 (x - 9)) (x + 2)}{(x + 4) (3 x^{3} - 10)}$

Step 8.3

Apply the distributive property.

$\frac{(x \cdot x + x \cdot - 9 - 5 x - 5 \cdot - 9) (x + 2)}{(x + 4) (3 x^{3} - 10)}$

Step 9

Simplify and combine like terms.

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Step 9.1

Simplify each term.

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Step 9.1.1

Multiply $x$ by $x$ .

$\frac{(x^{2} + x \cdot - 9 - 5 x - 5 \cdot - 9) (x + 2)}{(x + 4) (3 x^{3} - 10)}$

Step 9.1.2

Move $- 9$ to the left of $x$ .

$\frac{(x^{2} - 9 \cdot x - 5 x - 5 \cdot - 9) (x + 2)}{(x + 4) (3 x^{3} - 10)}$

Step 9.1.3

Multiply $- 5$ by $- 9$ .

$\frac{(x^{2} - 9 x - 5 x + 45) (x + 2)}{(x + 4) (3 x^{3} - 10)}$

Step 9.2

Subtract $5 x$ from $- 9 x$ .

$\frac{(x^{2} - 14 x + 45) (x + 2)}{(x + 4) (3 x^{3} - 10)}$

Step 10

Expand $(x^{2} - 14 x + 45) (x + 2)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{x^{2} x + x^{2} \cdot 2 - 14 x \cdot x - 14 x \cdot 2 + 45 x + 45 \cdot 2}{(x + 4) (3 x^{3} - 10)}$

Step 11

Simplify each term.

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Step 11.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 11.1.1

Multiply $x^{2}$ by $x$ .

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Step 11.1.1.1

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

$\frac{x^{2} x^{1} + x^{2} \cdot 2 - 14 x \cdot x - 14 x \cdot 2 + 45 x + 45 \cdot 2}{(x + 4) (3 x^{3} - 10)}$

Step 11.1.1.2

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

$\frac{x^{2 + 1} + x^{2} \cdot 2 - 14 x \cdot x - 14 x \cdot 2 + 45 x + 45 \cdot 2}{(x + 4) (3 x^{3} - 10)}$

Step 11.1.2

Add $2$ and $1$ .

$\frac{x^{3} + x^{2} \cdot 2 - 14 x \cdot x - 14 x \cdot 2 + 45 x + 45 \cdot 2}{(x + 4) (3 x^{3} - 10)}$

Step 11.2

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

$\frac{x^{3} + 2 \cdot x^{2} - 14 x \cdot x - 14 x \cdot 2 + 45 x + 45 \cdot 2}{(x + 4) (3 x^{3} - 10)}$

Step 11.3

Multiply $x$ by $x$ by adding the exponents.

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Step 11.3.1

Move $x$ .

$\frac{x^{3} + 2 x^{2} - 14 (x \cdot x) - 14 x \cdot 2 + 45 x + 45 \cdot 2}{(x + 4) (3 x^{3} - 10)}$

Step 11.3.2

Multiply $x$ by $x$ .

$\frac{x^{3} + 2 x^{2} - 14 x^{2} - 14 x \cdot 2 + 45 x + 45 \cdot 2}{(x + 4) (3 x^{3} - 10)}$

Step 11.4

Multiply $2$ by $- 14$ .

$\frac{x^{3} + 2 x^{2} - 14 x^{2} - 28 x + 45 x + 45 \cdot 2}{(x + 4) (3 x^{3} - 10)}$

Step 11.5

Multiply $45$ by $2$ .

$\frac{x^{3} + 2 x^{2} - 14 x^{2} - 28 x + 45 x + 90}{(x + 4) (3 x^{3} - 10)}$

Step 12

Subtract $14 x^{2}$ from $2 x^{2}$ .

$\frac{x^{3} - 12 x^{2} - 28 x + 45 x + 90}{(x + 4) (3 x^{3} - 10)}$

Step 13

Add $- 28 x$ and $45 x$ .

$\frac{x^{3} - 12 x^{2} + 17 x + 90}{(x + 4) (3 x^{3} - 10)}$

Step 14

Split the fraction $\frac{x^{3} - 12 x^{2} + 17 x + 90}{(x + 4) (3 x^{3} - 10)}$ into two fractions.

$\frac{x^{3} - 12 x^{2} + 17 x}{(x + 4) (3 x^{3} - 10)} + \frac{90}{(x + 4) (3 x^{3} - 10)}$

Step 15

Split the fraction $\frac{x^{3} - 12 x^{2} + 17 x}{(x + 4) (3 x^{3} - 10)}$ into two fractions.

$\frac{x^{3} - 12 x^{2}}{(x + 4) (3 x^{3} - 10)} + \frac{17 x}{(x + 4) (3 x^{3} - 10)} + \frac{90}{(x + 4) (3 x^{3} - 10)}$

Step 16

Split the fraction $\frac{x^{3} - 12 x^{2}}{(x + 4) (3 x^{3} - 10)}$ into two fractions.

$\frac{x^{3}}{(x + 4) (3 x^{3} - 10)} + \frac{- 12 x^{2}}{(x + 4) (3 x^{3} - 10)} + \frac{17 x}{(x + 4) (3 x^{3} - 10)} + \frac{90}{(x + 4) (3 x^{3} - 10)}$

Step 17

Move the negative in front of the fraction.

$\frac{x^{3}}{(x + 4) (3 x^{3} - 10)} - \frac{12 x^{2}}{(x + 4) (3 x^{3} - 10)} + \frac{17 x}{(x + 4) (3 x^{3} - 10)} + \frac{90}{(x + 4) (3 x^{3} - 10)}$