Solve by Factoring 40/(x-2)=1+42/(x+2)

$\frac{40}{x - 2} = 1 + \frac{42}{x + 2}$

Move all the expressions to the left side of the equation.

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Move $1$ to the left side of the equation by subtracting it from both sides.

$\frac{40}{x - 2} - 1 = \frac{42}{x + 2}$

Move $\frac{42}{x + 2}$ to the left side of the equation by subtracting it from both sides.

$\frac{40}{x - 2} - 1 - \frac{42}{x + 2} = 0$

Simplify $\frac{40}{x - 2} - 1 - \frac{42}{x + 2}$ .

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

$\frac{40}{x - 2} - 1 \cdot \frac{x - 2}{x - 2} - \frac{42}{x + 2} = 0$

Simplify terms.

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Combine $- 1$ and $\frac{x - 2}{x - 2}$ .

$\frac{40}{x - 2} + \frac{- (x - 2)}{x - 2} - \frac{42}{x + 2} = 0$

Combine the numerators over the common denominator.

$\frac{40 - (x - 2)}{x - 2} - \frac{42}{x + 2} = 0$

Simplify the numerator.

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Apply the distributive property.

$\frac{40 - x - - 2}{x - 2} - \frac{42}{x + 2} = 0$

Multiply $- 1$ by $- 2$ .

$\frac{40 - x + 2}{x - 2} - \frac{42}{x + 2} = 0$

Add $40$ and $2$ .

$\frac{- x + 42}{x - 2} - \frac{42}{x + 2} = 0$

To write $\frac{- x + 42}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{- x + 42}{x - 2} \cdot \frac{x + 2}{x + 2} - \frac{42}{x + 2} = 0$

To write $- \frac{42}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{- x + 42}{x - 2} \cdot \frac{x + 2}{x + 2} - \frac{42}{x + 2} \cdot \frac{x - 2}{x - 2} = 0$

Write each expression with a common denominator of $(x - 2) (x + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{- x + 42}{x - 2}$ and $\frac{x + 2}{x + 2}$ .

$\frac{(- x + 42) (x + 2)}{(x - 2) (x + 2)} - \frac{42}{x + 2} \cdot \frac{x - 2}{x - 2} = 0$

Multiply $\frac{42}{x + 2}$ and $\frac{x - 2}{x - 2}$ .

$\frac{(- x + 42) (x + 2)}{(x - 2) (x + 2)} - \frac{42 (x - 2)}{(x + 2) (x - 2)} = 0$

Reorder the factors of $(x - 2) (x + 2)$ .

$\frac{(- x + 42) (x + 2)}{(x + 2) (x - 2)} - \frac{42 (x - 2)}{(x + 2) (x - 2)} = 0$

Combine the numerators over the common denominator.

$\frac{(- x + 42) (x + 2) - 42 (x - 2)}{(x + 2) (x - 2)} = 0$

Simplify the numerator.

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Expand $(- x + 42) (x + 2)$ using the FOIL Method.

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Apply the distributive property.

$\frac{- x (x + 2) + 42 (x + 2) - 42 (x - 2)}{(x + 2) (x - 2)} = 0$

Apply the distributive property.

$\frac{- x \cdot x - x \cdot 2 + 42 (x + 2) - 42 (x - 2)}{(x + 2) (x - 2)} = 0$

Apply the distributive property.

$\frac{- x \cdot x - x \cdot 2 + 42 x + 42 \cdot 2 - 42 (x - 2)}{(x + 2) (x - 2)} = 0$

Simplify and combine like terms.

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Simplify each term.

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

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Move $x$ .

$\frac{- (x \cdot x) - x \cdot 2 + 42 x + 42 \cdot 2 - 42 (x - 2)}{(x + 2) (x - 2)} = 0$

Multiply $x$ by $x$ .

$\frac{- x^{2} - x \cdot 2 + 42 x + 42 \cdot 2 - 42 (x - 2)}{(x + 2) (x - 2)} = 0$

Multiply $2$ by $- 1$ .

$\frac{- x^{2} - 2 x + 42 x + 42 \cdot 2 - 42 (x - 2)}{(x + 2) (x - 2)} = 0$

Multiply $42$ by $2$ .

$\frac{- x^{2} - 2 x + 42 x + 84 - 42 (x - 2)}{(x + 2) (x - 2)} = 0$

Add $- 2 x$ and $42 x$ .

$\frac{- x^{2} + 40 x + 84 - 42 (x - 2)}{(x + 2) (x - 2)} = 0$

Apply the distributive property.

$\frac{- x^{2} + 40 x + 84 - 42 x - 42 \cdot - 2}{(x + 2) (x - 2)} = 0$

Multiply $- 42$ by $- 2$ .

$\frac{- x^{2} + 40 x + 84 - 42 x + 84}{(x + 2) (x - 2)} = 0$

Subtract $42 x$ from $40 x$ .

$\frac{- x^{2} - 2 x + 84 + 84}{(x + 2) (x - 2)} = 0$

Add $84$ and $84$ .

$\frac{- x^{2} - 2 x + 168}{(x + 2) (x - 2)} = 0$

Factor by grouping.

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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 $a \cdot c = - 1 \cdot 168 = - 168$ and whose sum is $b = - 2$ .

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

$\frac{- x^{2} - 2 (x) + 168}{(x + 2) (x - 2)} = 0$

Rewrite $- 2$ as $12$ plus $- 14$

$\frac{- x^{2} + (12 - 14) x + 168}{(x + 2) (x - 2)} = 0$

Apply the distributive property.

$\frac{- x^{2} + 12 x - 14 x + 168}{(x + 2) (x - 2)} = 0$

Factor out the greatest common factor from each group.

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Group the first two terms and the last two terms.

$\frac{(- x^{2} + 12 x) - 14 x + 168}{(x + 2) (x - 2)} = 0$

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

$\frac{x (- x + 12) + 14 (- x + 12)}{(x + 2) (x - 2)} = 0$

Factor the polynomial by factoring out the greatest common factor, $- x + 12$ .

$\frac{(- x + 12) (x + 14)}{(x + 2) (x - 2)} = 0$

Simplify with factoring out.

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

$\frac{(- (x) + 12) (x + 14)}{(x + 2) (x - 2)} = 0$

Rewrite $12$ as $- 1 (- 12)$ .

$\frac{(- (x) - 1 (- 12)) (x + 14)}{(x + 2) (x - 2)} = 0$

Factor $- 1$ out of $- (x) - 1 (- 12)$ .

$\frac{- (x - 12) (x + 14)}{(x + 2) (x - 2)} = 0$

Simplify the expression.

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Rewrite $- (x - 12)$ as $- 1 (x - 12)$ .

$\frac{- 1 (x - 12) (x + 14)}{(x + 2) (x - 2)} = 0$

Move the negative in front of the fraction.

$- \frac{(x - 12) (x + 14)}{(x + 2) (x - 2)} = 0$

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

$\frac{40}{x - 2} - 1 \cdot \frac{x - 2}{x - 2} - \frac{42}{x + 2} = 0$

Simplify terms.

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Combine $- 1$ and $\frac{x - 2}{x - 2}$ .

$\frac{40}{x - 2} + \frac{- (x - 2)}{x - 2} - \frac{42}{x + 2} = 0$

Combine the numerators over the common denominator.

$\frac{40 - (x - 2)}{x - 2} - \frac{42}{x + 2} = 0$

Simplify the numerator.

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Apply the distributive property.

$\frac{40 - x + 2}{x - 2} - \frac{42}{x + 2} = 0$

Multiply $- 1$ by $- 2$ .

$\frac{40 - x + 2}{x - 2} - \frac{42}{x + 2} = 0$

Add $40$ and $2$ .

$\frac{- x + 42}{x - 2} - \frac{42}{x + 2} = 0$

To write $\frac{- x + 42}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{- x + 42}{x - 2} \cdot \frac{x + 2}{x + 2} - \frac{42}{x + 2} = 0$

To write $- \frac{42}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{- x + 42}{x - 2} \cdot \frac{x + 2}{x + 2} - \frac{42}{x + 2} \cdot \frac{x - 2}{x - 2} = 0$

Write each expression with a common denominator of $(x - 2) (x + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{- x + 42}{x - 2}$ and $\frac{x + 2}{x + 2}$ .

$\frac{(- x + 42) (x + 2)}{(x - 2) (x + 2)} - \frac{42}{x + 2} \cdot \frac{x - 2}{x - 2} = 0$

Multiply $\frac{42}{x + 2}$ and $\frac{x - 2}{x - 2}$ .

$\frac{(- x + 42) (x + 2)}{(x - 2) (x + 2)} - \frac{42 (x - 2)}{(x + 2) (x - 2)} = 0$

Reorder the factors of $(x - 2) (x + 2)$ .

$\frac{(- x + 42) (x + 2)}{(x + 2) (x - 2)} - \frac{42 (x - 2)}{(x + 2) (x - 2)} = 0$

Combine the numerators over the common denominator.

$\frac{(- x + 42) (x + 2) - 42 (x - 2)}{(x + 2) (x - 2)} = 0$

Simplify the numerator.

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Expand $(- x + 42) (x + 2)$ using the FOIL Method.

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Apply the distributive property.

$\frac{- x (x + 2) + 42 (x + 2) - 42 (x - 2)}{(x + 2) (x - 2)} = 0$

Apply the distributive property.

$\frac{- x \cdot x - x \cdot 2 + 42 (x + 2) - 42 (x - 2)}{(x + 2) (x - 2)} = 0$

Apply the distributive property.

$\frac{- x \cdot x - x \cdot 2 + 42 x + 42 \cdot 2 - 42 (x - 2)}{(x + 2) (x - 2)} = 0$

Simplify and combine like terms.

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Simplify each term.

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

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Move $x$ .

$\frac{- (x \cdot x) - x \cdot 2 + 42 x + 42 \cdot 2 - 42 (x - 2)}{(x + 2) (x - 2)} = 0$

Multiply $x$ by $x$ .

$\frac{- x^{2} - x \cdot 2 + 42 x + 42 \cdot 2 - 42 (x - 2)}{(x + 2) (x - 2)} = 0$

Multiply $2$ by $- 1$ .

$\frac{- x^{2} - 2 x + 42 x + 42 \cdot 2 - 42 (x - 2)}{(x + 2) (x - 2)} = 0$

Multiply $42$ by $2$ .

$\frac{- x^{2} - 2 x + 42 x + 84 - 42 (x - 2)}{(x + 2) (x - 2)} = 0$

Add $- 2 x$ and $42 x$ .

$\frac{- x^{2} + 40 x + 84 - 42 (x - 2)}{(x + 2) (x - 2)} = 0$

Apply the distributive property.

$\frac{- x^{2} + 40 x + 84 - 42 x - 42 \cdot - 2}{(x + 2) (x - 2)} = 0$

Multiply $- 42$ by $- 2$ .

$\frac{- x^{2} + 40 x + 84 - 42 x + 84}{(x + 2) (x - 2)} = 0$

Subtract $42 x$ from $40 x$ .

$\frac{- x^{2} - 2 x + 84 + 84}{(x + 2) (x - 2)} = 0$

Add $84$ and $84$ .

$\frac{- x^{2} - 2 x + 168}{(x + 2) (x - 2)} = 0$

Factor by grouping.

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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 $a \cdot c = - 1 \cdot 168 = - 168$ and whose sum is $b = - 2$ .

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

$\frac{- x^{2} - 2 x + 168}{(x + 2) (x - 2)} = 0$

Rewrite $- 2$ as $12$ plus $- 14$

$\frac{- x^{2} + (12 - 14) x + 168}{(x + 2) (x - 2)} = 0$

Apply the distributive property.

$\frac{- x^{2} + 12 x - 14 x + 168}{(x + 2) (x - 2)} = 0$

Factor out the greatest common factor from each group.

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Group the first two terms and the last two terms.

$\frac{(- x^{2} + 12 x) - 14 x + 168}{(x + 2) (x - 2)} = 0$

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

$\frac{x (- x + 12) + 14 (- x + 12)}{(x + 2) (x - 2)} = 0$

Factor the polynomial by factoring out the greatest common factor, $- x + 12$ .

$\frac{(- x + 12) (x + 14)}{(x + 2) (x - 2)} = 0$

Simplify with factoring out.

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

$\frac{(- (x) + 12) (x + 14)}{(x + 2) (x - 2)} = 0$

Rewrite $12$ as $- 1 (- 12)$ .

$\frac{(- (x) - 1 \cdot - 12) (x + 14)}{(x + 2) (x - 2)} = 0$

Factor $- 1$ out of $- (x) - 1 (- 12)$ .

$\frac{- (x - 12) (x + 14)}{(x + 2) (x - 2)} = 0$

Simplify the expression.

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Rewrite $- (x - 12)$ as $- 1 (x - 12)$ .

$\frac{- 1 (x - 12) (x + 14)}{(x + 2) (x - 2)} = 0$

Move the negative in front of the fraction.

$- \frac{(x - 12) (x + 14)}{(x + 2) (x - 2)} = 0$

Find the LCD of the terms in the equation.

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Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$(x + 2) (x - 2), 1$

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

The LCM of $1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

The factor for $x + 2$ is $x + 2$ itself.

$(x + 2) = x + 2$

$(x + 2)$ occurs $1$ time.

The factor for $x - 2$ is $x - 2$ itself.

$(x - 2) = x - 2$

$(x - 2)$ occurs $1$ time.

The LCM of $x + 2, x - 2$ is the result of multiplying all factors the greatest number of times they occur in either term.

$(x + 2) (x - 2)$

Multiply each term by $(x + 2) (x - 2)$ and simplify.

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Multiply each term in $- \frac{(x - 12) (x + 14)}{(x + 2) (x - 2)} = 0$ by $(x + 2) (x - 2)$ in order to remove all the denominators from the equation.

$- \frac{(x - 12) (x + 14)}{(x + 2) (x - 2)} \cdot ((x + 2) (x - 2)) = 0 \cdot ((x + 2) (x - 2))$

Simplify $- \frac{(x - 12) (x + 14)}{(x + 2) (x - 2)} \cdot ((x + 2) (x - 2))$ .

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Simplify terms.

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Cancel the common factor of $(x + 2) (x - 2)$ .

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Move the leading negative in $- \frac{(x - 12) (x + 14)}{(x + 2) (x - 2)}$ into the numerator.

$\frac{- (x - 12) (x + 14)}{(x + 2) (x - 2)} \cdot ((x + 2) (x - 2)) = 0 \cdot ((x + 2) (x - 2))$

Cancel the common factor.

$\frac{- (x - 12) (x + 14)}{(x + 2) (x - 2)} \cdot ((x + 2) (x - 2)) = 0 \cdot ((x + 2) (x - 2))$

Rewrite the expression.

$- (x - 12) (x + 14) = 0 \cdot ((x + 2) (x - 2))$

Apply the distributive property.

$(- x - - 12) (x + 14) = 0 \cdot ((x + 2) (x - 2))$

Multiply $- 1$ by $- 12$ .

$(- x + 12) (x + 14) = 0 \cdot ((x + 2) (x - 2))$

Expand $(- x + 12) (x + 14)$ using the FOIL Method.

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Apply the distributive property.

$- x (x + 14) + 12 (x + 14) = 0 \cdot ((x + 2) (x - 2))$

Apply the distributive property.

$- x \cdot x - x \cdot 14 + 12 (x + 14) = 0 \cdot ((x + 2) (x - 2))$

Apply the distributive property.

$- x \cdot x - x \cdot 14 + 12 x + 12 \cdot 14 = 0 \cdot ((x + 2) (x - 2))$

Simplify and combine like terms.

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Simplify each term.

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

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Move $x$ .

$- (x \cdot x) - x \cdot 14 + 12 x + 12 \cdot 14 = 0 \cdot ((x + 2) (x - 2))$

Multiply $x$ by $x$ .

$- x^{2} - x \cdot 14 + 12 x + 12 \cdot 14 = 0 \cdot ((x + 2) (x - 2))$

Multiply $14$ by $- 1$ .

$- x^{2} - 14 x + 12 x + 12 \cdot 14 = 0 \cdot ((x + 2) (x - 2))$

Multiply $12$ by $14$ .

$- x^{2} - 14 x + 12 x + 168 = 0 \cdot ((x + 2) (x - 2))$

Add $- 14 x$ and $12 x$ .

$- x^{2} - 2 x + 168 = 0 \cdot ((x + 2) (x - 2))$

Simplify $0 \cdot ((x + 2) (x - 2))$ .

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

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Apply the distributive property.

$- x^{2} - 2 x + 168 = 0 \cdot (x (x - 2) + 2 (x - 2))$

Apply the distributive property.

$- x^{2} - 2 x + 168 = 0 \cdot (x \cdot x + x \cdot - 2 + 2 (x - 2))$

Apply the distributive property.

$- x^{2} - 2 x + 168 = 0 \cdot (x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2)$

Simplify terms.

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Combine the opposite terms in $x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2$ .

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Reorder the factors in the terms $x \cdot - 2$ and $2 x$ .

$- x^{2} - 2 x + 168 = 0 \cdot (x \cdot x - 2 x + 2 x + 2 \cdot - 2)$

Add $- 2 x$ and $2 x$ .

$- x^{2} - 2 x + 168 = 0 \cdot (x \cdot x + 0 + 2 \cdot - 2)$

Add $x \cdot x$ and $0$ .

$- x^{2} - 2 x + 168 = 0 \cdot (x \cdot x + 2 \cdot - 2)$

Simplify each term.

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Multiply $x$ by $x$ .

$- x^{2} - 2 x + 168 = 0 \cdot (x^{2} + 2 \cdot - 2)$

Multiply $2$ by $- 2$ .

$- x^{2} - 2 x + 168 = 0 \cdot (x^{2} - 4)$

Multiply $0$ by $x^{2} - 4$ .

$- x^{2} - 2 x + 168 = 0$

Solve the equation.

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Factor the left side of the equation.

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Factor $- 1$ out of $- x^{2} - 2 x + 168$ .

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Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) - 2 x + 168 = 0$

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

$- (x^{2}) - (2 x) + 168 = 0$

Rewrite $168$ as $- 1 (- 168)$ .

$- (x^{2}) - (2 x) - 1 \cdot - 168 = 0$

Factor $- 1$ out of $- (x^{2}) - (2 x)$ .

$- (x^{2} + 2 x) - 1 \cdot - 168 = 0$

Factor $- 1$ out of $- (x^{2} + 2 x) - 1 (- 168)$ .

$- (x^{2} + 2 x - 168) = 0$

Factor.

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Factor $x^{2} + 2 x - 168$ using the AC method.

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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 $- 168$ and whose sum is $2$ .

$- 12, 14$

Write the factored form using these integers.

$- ((x - 12) (x + 14)) = 0$

Remove unnecessary parentheses.

$- (x - 12) (x + 14) = 0$

Multiply each term in $- (x - 12) (x + 14) = 0$ by $- 1$

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Multiply each term in $- (x - 12) (x + 14) = 0$ by $- 1$ .

$(- (x - 12) (x + 14)) \cdot - 1 = 0 \cdot - 1$

Simplify $(- (x - 12) (x + 14)) \cdot - 1$ .

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Simplify by multiplying through.

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Apply the distributive property.

$(- x - - 12) (x + 14) \cdot - 1 = 0 \cdot - 1$

Multiply $- 1$ by $- 12$ .

$(- x + 12) (x + 14) \cdot - 1 = 0 \cdot - 1$

Expand $(- x + 12) (x + 14)$ using the FOIL Method.

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Apply the distributive property.

$(- x (x + 14) + 12 (x + 14)) \cdot - 1 = 0 \cdot - 1$

Apply the distributive property.

$(- x \cdot x - x \cdot 14 + 12 (x + 14)) \cdot - 1 = 0 \cdot - 1$

Apply the distributive property.

$(- x \cdot x - x \cdot 14 + 12 x + 12 \cdot 14) \cdot - 1 = 0 \cdot - 1$

Simplify and combine like terms.

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Simplify each term.

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

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Move $x$ .

$(- (x \cdot x) - x \cdot 14 + 12 x + 12 \cdot 14) \cdot - 1 = 0 \cdot - 1$

Multiply $x$ by $x$ .

$(- x^{2} - x \cdot 14 + 12 x + 12 \cdot 14) \cdot - 1 = 0 \cdot - 1$

Multiply $14$ by $- 1$ .

$(- x^{2} - 14 x + 12 x + 12 \cdot 14) \cdot - 1 = 0 \cdot - 1$

Multiply $12$ by $14$ .

$(- x^{2} - 14 x + 12 x + 168) \cdot - 1 = 0 \cdot - 1$

Add $- 14 x$ and $12 x$ .

$(- x^{2} - 2 x + 168) \cdot - 1 = 0 \cdot - 1$

Apply the distributive property.

$- x^{2} \cdot - 1 - 2 x \cdot - 1 + 168 \cdot - 1 = 0 \cdot - 1$

Simplify.

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Multiply $- x^{2} \cdot - 1$ .

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Multiply $- 1$ by $- 1$ .

$1 x^{2} - 2 x \cdot - 1 + 168 \cdot - 1 = 0 \cdot - 1$

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

$x^{2} - 2 x \cdot - 1 + 168 \cdot - 1 = 0 \cdot - 1$

Multiply $- 1$ by $- 2$ .

$x^{2} + 2 x + 168 \cdot - 1 = 0 \cdot - 1$

Multiply $168$ by $- 1$ .

$x^{2} + 2 x - 168 = 0 \cdot - 1$

Multiply $0$ by $- 1$ .

$x^{2} + 2 x - 168 = 0$

Factor $x^{2} + 2 x - 168$ using the AC method.

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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 $- 168$ and whose sum is $2$ .

$- 12, 14$

Write the factored form using these integers.

$(x - 12) (x + 14) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 12 = 0$

$x + 14 = 0$

Set the first factor equal to $0$ and solve.

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Set the first factor equal to $0$ .

$x - 12 = 0$

Add $12$ to both sides of the equation.

$x = 12$

Set the next factor equal to $0$ and solve.

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Set the next factor equal to $0$ .

$x + 14 = 0$

Subtract $14$ from both sides of the equation.

$x = - 14$

The final solution is all the values that make $(x - 12) (x + 14) = 0$ true.

$x = 12, - 14$

Solve by Factoring 40/(x-2)=1+42/(x+2)

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