Pre-Algebra Examples

$\frac{\frac{2 x^{2} - 10 x + 12}{x^{2} - 7 x + 12}}{\frac{2 x^{2} \cdot (16 x) + 24}{x^{2} - 6 x + 8}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 x^{2} - 10 x + 12}{x^{2} - 7 x + 12} \cdot \frac{x^{2} - 6 x + 8}{2 x^{2} \cdot (16 x) + 24}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $2$ out of $2 x^{2} - 10 x + 12$ .

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Step 2.1.1

Factor $2$ out of $2 x^{2}$ .

$\frac{2 (x^{2}) - 10 x + 12}{x^{2} - 7 x + 12} \cdot \frac{x^{2} - 6 x + 8}{2 x^{2} \cdot (16 x) + 24}$

Step 2.1.2

Factor $2$ out of $- 10 x$ .

$\frac{2 (x^{2}) + 2 (- 5 x) + 12}{x^{2} - 7 x + 12} \cdot \frac{x^{2} - 6 x + 8}{2 x^{2} \cdot (16 x) + 24}$

Step 2.1.3

Factor $2$ out of $12$ .

$\frac{2 x^{2} + 2 (- 5 x) + 2 \cdot 6}{x^{2} - 7 x + 12} \cdot \frac{x^{2} - 6 x + 8}{2 x^{2} \cdot (16 x) + 24}$

Step 2.1.4

Factor $2$ out of $2 x^{2} + 2 (- 5 x)$ .

$\frac{2 (x^{2} - 5 x) + 2 \cdot 6}{x^{2} - 7 x + 12} \cdot \frac{x^{2} - 6 x + 8}{2 x^{2} \cdot (16 x) + 24}$

Step 2.1.5

Factor $2$ out of $2 (x^{2} - 5 x) + 2 \cdot 6$ .

$\frac{2 (x^{2} - 5 x + 6)}{x^{2} - 7 x + 12} \cdot \frac{x^{2} - 6 x + 8}{2 x^{2} \cdot (16 x) + 24}$

Step 2.2

Factor $x^{2} - 5 x + 6$ using the AC method.

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Step 2.2.1

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 $6$ and whose sum is $- 5$ .

$- 3, - 2$

Step 2.2.2

Write the factored form using these integers.

$\frac{2 ((x - 3) (x - 2))}{x^{2} - 7 x + 12} \cdot \frac{x^{2} - 6 x + 8}{2 x^{2} \cdot (16 x) + 24}$

$\frac{2 (x - 3) (x - 2)}{x^{2} - 7 x + 12} \cdot \frac{x^{2} - 6 x + 8}{2 x^{2} \cdot (16 x) + 24}$

Step 3

Factor $x^{2} - 7 x + 12$ 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 $12$ and whose sum is $- 7$ .

$- 4, - 3$

Step 3.2

Write the factored form using these integers.

$\frac{2 (x - 3) (x - 2)}{(x - 4) (x - 3)} \cdot \frac{x^{2} - 6 x + 8}{2 x^{2} \cdot (16 x) + 24}$

Step 4

Cancel the common factor of $x - 3$ .

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Step 4.1

Cancel the common factor.

$\frac{2 (x - 3) (x - 2)}{(x - 4) (x - 3)} \cdot \frac{x^{2} - 6 x + 8}{2 x^{2} \cdot (16 x) + 24}$

Step 4.2

Rewrite the expression.

$\frac{2 (x - 2)}{x - 4} \cdot \frac{x^{2} - 6 x + 8}{2 x^{2} \cdot (16 x) + 24}$

Step 5

Factor $x^{2} - 6 x + 8$ 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 $8$ and whose sum is $- 6$ .

$- 4, - 2$

Step 5.2

Write the factored form using these integers.

$\frac{2 (x - 2)}{x - 4} \cdot \frac{(x - 4) (x - 2)}{2 x^{2} \cdot (16 x) + 24}$

Step 6

Simplify the denominator.

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Step 6.1

Rewrite using the commutative property of multiplication.

$\frac{2 (x - 2)}{x - 4} \cdot \frac{(x - 4) (x - 2)}{2 \cdot 16 x^{2} x + 24}$

Step 6.2

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

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Step 6.2.1

Move $x$ .

$\frac{2 (x - 2)}{x - 4} \cdot \frac{(x - 4) (x - 2)}{2 \cdot 16 (x \cdot x^{2}) + 24}$

Step 6.2.2

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

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Step 6.2.2.1

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

$\frac{2 (x - 2)}{x - 4} \cdot \frac{(x - 4) (x - 2)}{2 \cdot 16 (x^{1} x^{2}) + 24}$

Step 6.2.2.2

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

$\frac{2 (x - 2)}{x - 4} \cdot \frac{(x - 4) (x - 2)}{2 \cdot 16 x^{1 + 2} + 24}$

Step 6.2.3

Add $1$ and $2$ .

$\frac{2 (x - 2)}{x - 4} \cdot \frac{(x - 4) (x - 2)}{2 \cdot 16 x^{3} + 24}$

Step 6.3

Multiply $2$ by $16$ .

$\frac{2 (x - 2)}{x - 4} \cdot \frac{(x - 4) (x - 2)}{32 x^{3} + 24}$

Step 6.4

Factor $8$ out of $32 x^{3} + 24$ .

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Step 6.4.1

Factor $8$ out of $32 x^{3}$ .

$\frac{2 (x - 2)}{x - 4} \cdot \frac{(x - 4) (x - 2)}{8 (4 x^{3}) + 24}$

Step 6.4.2

Factor $8$ out of $24$ .

$\frac{2 (x - 2)}{x - 4} \cdot \frac{(x - 4) (x - 2)}{8 (4 x^{3}) + 8 (3)}$

Step 6.4.3

Factor $8$ out of $8 (4 x^{3}) + 8 (3)$ .

$\frac{2 (x - 2)}{x - 4} \cdot \frac{(x - 4) (x - 2)}{8 (4 x^{3} + 3)}$

Step 7

Cancel the common factor of $2$ .

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Step 7.1

Factor $2$ out of $8 (4 x^{3} + 3)$ .

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

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

Step 8

Cancel the common factor of $x - 4$ .

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Step 8.1

Cancel the common factor.

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

Step 8.2

Rewrite the expression.

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

Step 9

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

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

Step 10

Simplify the numerator.

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Step 10.1

Raise $x - 2$ to the power of $1$ .

$\frac{{(x - 2)}^{1} (x - 2)}{4 (4 x^{3} + 3)}$

Step 10.2

Raise $x - 2$ to the power of $1$ .

$\frac{{(x - 2)}^{1} {(x - 2)}^{1}}{4 (4 x^{3} + 3)}$

Step 10.3

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

$\frac{{(x - 2)}^{1 + 1}}{4 (4 x^{3} + 3)}$

Step 10.4

Add $1$ and $1$ .

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

Step 11

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

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

Step 12

Expand $(x - 2) (x - 2)$ using the FOIL Method.

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Step 12.1

Apply the distributive property.

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

Step 12.2

Apply the distributive property.

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

Step 12.3

Apply the distributive property.

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

Step 13

Simplify and combine like terms.

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Step 13.1

Simplify each term.

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Step 13.1.1

Multiply $x$ by $x$ .

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

Step 13.1.2

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

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

Step 13.1.3

Multiply $- 2$ by $- 2$ .

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

Step 13.2

Subtract $2 x$ from $- 2 x$ .

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

Step 14

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

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

Step 15

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

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

Step 16

Cancel the common factor of $- 4$ and $4$ .

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Step 16.1

Factor $4$ out of $- 4 x$ .

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

Step 16.2

Cancel the common factors.

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Step 16.2.1

Cancel the common factor.

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

Step 16.2.2

Rewrite the expression.

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

Step 17

Move the negative in front of the fraction.

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

Step 18

Cancel the common factor of $4$ .

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Step 18.1

Cancel the common factor.

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

Step 18.2

Rewrite the expression.

$\frac{x^{2}}{4 (4 x^{3} + 3)} - \frac{x}{4 x^{3} + 3} + \frac{1}{4 x^{3} + 3}$