Pre-Algebra Examples

$\frac{x - 2}{x - 8} - \frac{3 x}{4 - x} + \frac{x^{2} - 88}{x^{2} - 12 x + 32}$

Step 1

Factor $x^{2} - 12 x + 32$ using the AC method.

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Step 1.1

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 $32$ and whose sum is $- 12$ .

$- 8, - 4$

Step 1.2

Write the factored form using these integers.

$\frac{x - 2}{x - 8} - \frac{3 x}{4 - x} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 2

To write $\frac{x - 2}{x - 8}$ as a fraction with a common denominator, multiply by $\frac{4 - x}{4 - x}$ .

$\frac{x - 2}{x - 8} \cdot \frac{4 - x}{4 - x} - \frac{3 x}{4 - x} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 3

To write $- \frac{3 x}{4 - x}$ as a fraction with a common denominator, multiply by $\frac{x - 8}{x - 8}$ .

$\frac{x - 2}{x - 8} \cdot \frac{4 - x}{4 - x} - \frac{3 x}{4 - x} \cdot \frac{x - 8}{x - 8} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 4

Write each expression with a common denominator of $(x - 8) (4 - x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{x - 2}{x - 8}$ by $\frac{4 - x}{4 - x}$ .

$\frac{(x - 2) (4 - x)}{(x - 8) (4 - x)} - \frac{3 x}{4 - x} \cdot \frac{x - 8}{x - 8} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 4.2

Multiply $\frac{3 x}{4 - x}$ by $\frac{x - 8}{x - 8}$ .

$\frac{(x - 2) (4 - x)}{(x - 8) (4 - x)} - \frac{3 x (x - 8)}{(4 - x) (x - 8)} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 4.3

Reorder the factors of $(x - 8) (4 - x)$ .

$\frac{(x - 2) (4 - x)}{(4 - x) (x - 8)} - \frac{3 x (x - 8)}{(4 - x) (x - 8)} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(x - 2) (4 - x) - 3 x (x - 8)}{(4 - x) (x - 8)} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 6

Simplify the numerator.

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Step 6.1

Expand $(x - 2) (4 - x)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$\frac{x (4 - x) - 2 (4 - x) - 3 x (x - 8)}{(4 - x) (x - 8)} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 6.1.2

Apply the distributive property.

$\frac{x \cdot 4 + x (- x) - 2 (4 - x) - 3 x (x - 8)}{(4 - x) (x - 8)} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 6.1.3

Apply the distributive property.

$\frac{x \cdot 4 + x (- x) - 2 \cdot 4 - 2 (- x) - 3 x (x - 8)}{(4 - x) (x - 8)} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Move $4$ to the left of $x$ .

$\frac{4 \cdot x + x (- x) - 2 \cdot 4 - 2 (- x) - 3 x (x - 8)}{(4 - x) (x - 8)} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 6.2.1.2

Rewrite using the commutative property of multiplication.

$\frac{4 x - x \cdot x - 2 \cdot 4 - 2 (- x) - 3 x (x - 8)}{(4 - x) (x - 8)} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 6.2.1.3

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

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Step 6.2.1.3.1

Move $x$ .

$\frac{4 x - (x \cdot x) - 2 \cdot 4 - 2 (- x) - 3 x (x - 8)}{(4 - x) (x - 8)} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 6.2.1.3.2

Multiply $x$ by $x$ .

$\frac{4 x - x^{2} - 2 \cdot 4 - 2 (- x) - 3 x (x - 8)}{(4 - x) (x - 8)} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 6.2.1.4

Multiply $- 2$ by $4$ .

$\frac{4 x - x^{2} - 8 - 2 (- x) - 3 x (x - 8)}{(4 - x) (x - 8)} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 6.2.1.5

Multiply $- 1$ by $- 2$ .

$\frac{4 x - x^{2} - 8 + 2 x - 3 x (x - 8)}{(4 - x) (x - 8)} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 6.2.2

Add $4 x$ and $2 x$ .

$\frac{6 x - x^{2} - 8 - 3 x (x - 8)}{(4 - x) (x - 8)} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 6.3

Apply the distributive property.

$\frac{6 x - x^{2} - 8 - 3 x \cdot x - 3 x \cdot - 8}{(4 - x) (x - 8)} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 6.4

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

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Step 6.4.1

Move $x$ .

$\frac{6 x - x^{2} - 8 - 3 (x \cdot x) - 3 x \cdot - 8}{(4 - x) (x - 8)} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 6.4.2

Multiply $x$ by $x$ .

$\frac{6 x - x^{2} - 8 - 3 x^{2} - 3 x \cdot - 8}{(4 - x) (x - 8)} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 6.5

Multiply $- 8$ by $- 3$ .

$\frac{6 x - x^{2} - 8 - 3 x^{2} + 24 x}{(4 - x) (x - 8)} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 6.6

Add $6 x$ and $24 x$ .

$\frac{- x^{2} - 8 - 3 x^{2} + 30 x}{(4 - x) (x - 8)} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 6.7

Subtract $3 x^{2}$ from $- x^{2}$ .

$\frac{- 4 x^{2} - 8 + 30 x}{(4 - x) (x - 8)} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 6.8

Reorder terms.

$\frac{- 4 x^{2} + 30 x - 8}{(4 - x) (x - 8)} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 6.9

Factor $2$ out of $- 4 x^{2} + 30 x - 8$ .

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Step 6.9.1

Factor $2$ out of $- 4 x^{2}$ .

$\frac{2 (- 2 x^{2}) + 30 x - 8}{(4 - x) (x - 8)} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 6.9.2

Factor $2$ out of $30 x$ .

$\frac{2 (- 2 x^{2}) + 2 (15 x) - 8}{(4 - x) (x - 8)} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 6.9.3

Factor $2$ out of $- 8$ .

$\frac{2 (- 2 x^{2}) + 2 (15 x) + 2 (- 4)}{(4 - x) (x - 8)} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 6.9.4

Factor $2$ out of $2 (- 2 x^{2}) + 2 (15 x)$ .

$\frac{2 (- 2 x^{2} + 15 x) + 2 (- 4)}{(4 - x) (x - 8)} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 6.9.5

Factor $2$ out of $2 (- 2 x^{2} + 15 x) + 2 (- 4)$ .

$\frac{2 (- 2 x^{2} + 15 x - 4)}{(4 - x) (x - 8)} + \frac{x^{2} - 88}{(x - 8) (x - 4)}$

Step 7

Simplify terms.

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Step 7.1

Factor $- 1$ out of $x$ .

$\frac{2 (- 2 x^{2} + 15 x - 4)}{(4 - x) (x - 8)} + \frac{x^{2} - 88}{(x - 8) (- 1 (- x) - 4)}$

Step 7.2

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{2 (- 2 x^{2} + 15 x - 4)}{(4 - x) (x - 8)} + \frac{x^{2} - 88}{(x - 8) (- 1 (- x) - 1 (4))}$

Step 7.3

Factor $- 1$ out of $- 1 (- x) - 1 (4)$ .

$\frac{2 (- 2 x^{2} + 15 x - 4)}{(4 - x) (x - 8)} + \frac{x^{2} - 88}{(x - 8) (- 1 (- x + 4))}$

Step 7.4

Simplify the expression.

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Step 7.4.1

Move a negative from the denominator of $\frac{x^{2} - 88}{(x - 8) (- 1 (- x + 4))}$ to the numerator.

$\frac{2 (- 2 x^{2} + 15 x - 4)}{(4 - x) (x - 8)} + \frac{- (x^{2} - 88)}{(x - 8) (- x + 4)}$

Step 7.4.2

Reorder terms.

$\frac{2 (- 2 x^{2} + 15 x - 4)}{(- x + 4) (x - 8)} + \frac{- (x^{2} - 88)}{(x - 8) (- x + 4)}$

Step 7.4.3

Reorder the factors of $(x - 8) (- x + 4)$ .

$\frac{2 (- 2 x^{2} + 15 x - 4)}{(- x + 4) (x - 8)} + \frac{- (x^{2} - 88)}{(- x + 4) (x - 8)}$

Step 7.5

Combine the numerators over the common denominator.

$\frac{2 (- 2 x^{2} + 15 x - 4) - (x^{2} - 88)}{(- x + 4) (x - 8)}$

Step 8

Simplify the numerator.

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Step 8.1

Apply the distributive property.

$\frac{2 (- 2 x^{2}) + 2 (15 x) + 2 \cdot - 4 - (x^{2} - 88)}{(- x + 4) (x - 8)}$

Step 8.2

Simplify.

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Step 8.2.1

Multiply $- 2$ by $2$ .

$\frac{- 4 x^{2} + 2 (15 x) + 2 \cdot - 4 - (x^{2} - 88)}{(- x + 4) (x - 8)}$

Step 8.2.2

Multiply $15$ by $2$ .

$\frac{- 4 x^{2} + 30 x + 2 \cdot - 4 - (x^{2} - 88)}{(- x + 4) (x - 8)}$

Step 8.2.3

Multiply $2$ by $- 4$ .

$\frac{- 4 x^{2} + 30 x - 8 - (x^{2} - 88)}{(- x + 4) (x - 8)}$

Step 8.3

Apply the distributive property.

$\frac{- 4 x^{2} + 30 x - 8 - x^{2} - - 88}{(- x + 4) (x - 8)}$

Step 8.4

Multiply $- 1$ by $- 88$ .

$\frac{- 4 x^{2} + 30 x - 8 - x^{2} + 88}{(- x + 4) (x - 8)}$

Step 8.5

Subtract $x^{2}$ from $- 4 x^{2}$ .

$\frac{- 5 x^{2} + 30 x - 8 + 88}{(- x + 4) (x - 8)}$

Step 8.6

Add $- 8$ and $88$ .

$\frac{- 5 x^{2} + 30 x + 80}{(- x + 4) (x - 8)}$

Step 8.7

Rewrite $- 5 x^{2} + 30 x + 80$ in a factored form.

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Step 8.7.1

Factor $5$ out of $- 5 x^{2} + 30 x + 80$ .

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Step 8.7.1.1

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

$\frac{5 (- x^{2}) + 30 x + 80}{(- x + 4) (x - 8)}$

Step 8.7.1.2

Factor $5$ out of $30 x$ .

$\frac{5 (- x^{2}) + 5 (6 x) + 80}{(- x + 4) (x - 8)}$

Step 8.7.1.3

Factor $5$ out of $80$ .

$\frac{5 (- x^{2}) + 5 (6 x) + 5 (16)}{(- x + 4) (x - 8)}$

Step 8.7.1.4

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

$\frac{5 (- x^{2} + 6 x) + 5 (16)}{(- x + 4) (x - 8)}$

Step 8.7.1.5

Factor $5$ out of $5 (- x^{2} + 6 x) + 5 (16)$ .

$\frac{5 (- x^{2} + 6 x + 16)}{(- x + 4) (x - 8)}$

Step 8.7.2

Factor by grouping.

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Step 8.7.2.1

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 $a \cdot c = - 1 \cdot 16 = - 16$ and whose sum is $b = 6$ .

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Step 8.7.2.1.1

Factor $6$ out of $6 x$ .

$\frac{5 (- x^{2} + 6 (x) + 16)}{(- x + 4) (x - 8)}$

Step 8.7.2.1.2

Rewrite $6$ as $- 2$ plus $8$

$\frac{5 (- x^{2} + (- 2 + 8) x + 16)}{(- x + 4) (x - 8)}$

Step 8.7.2.1.3

Apply the distributive property.

$\frac{5 (- x^{2} - 2 x + 8 x + 16)}{(- x + 4) (x - 8)}$

Step 8.7.2.2

Factor out the greatest common factor from each group.

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Step 8.7.2.2.1

Group the first two terms and the last two terms.

$\frac{5 ((- x^{2} - 2 x) + 8 x + 16)}{(- x + 4) (x - 8)}$

Step 8.7.2.2.2

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

$\frac{5 (x (- x - 2) - 8 (- x - 2))}{(- x + 4) (x - 8)}$

Step 8.7.2.3

Factor the polynomial by factoring out the greatest common factor, $- x - 2$ .

$\frac{5 ((- x - 2) (x - 8))}{(- x + 4) (x - 8)}$

$\frac{5 (- x - 2) (x - 8)}{(- x + 4) (x - 8)}$

Step 9

Simplify terms.

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Step 9.1

Cancel the common factor of $x - 8$ .

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Step 9.1.1

Cancel the common factor.

$\frac{5 (- x - 2) (x - 8)}{(- x + 4) (x - 8)}$

Step 9.1.2

Rewrite the expression.

$\frac{5 (- x - 2)}{- x + 4}$

Step 9.2

Factor $- 1$ out of $- x$ .

$\frac{5 (- (x) - 2)}{- x + 4}$

Step 9.3

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{5 (- (x) - 1 (2))}{- x + 4}$

Step 9.4

Factor $- 1$ out of $- (x) - 1 (2)$ .

$\frac{5 (- (x + 2))}{- x + 4}$

Step 9.5

Rewrite $- (x + 2)$ as $- 1 (x + 2)$ .

$\frac{5 (- 1 (x + 2))}{- x + 4}$

Step 9.6

Factor $- 1$ out of $- x$ .

$\frac{5 (- 1 (x + 2))}{- (x) + 4}$

Step 9.7

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{5 (- 1 (x + 2))}{- (x) - 1 (- 4)}$

Step 9.8

Factor $- 1$ out of $- (x) - 1 (- 4)$ .

$\frac{5 (- 1 (x + 2))}{- (x - 4)}$

Step 9.9

Rewrite $- (x - 4)$ as $- 1 (x - 4)$ .

$\frac{5 (- 1 (x + 2))}{- 1 (x - 4)}$

Step 9.10

Cancel the common factor.

$\frac{5 (- 1 (x + 2))}{- 1 (x - 4)}$

Step 9.11

Rewrite the expression.

$\frac{5 (x + 2)}{x - 4}$