Solve by Factoring (x+1)/(x+6)=(x^2-38x)/(x^2+4x-12)-(x-6)/(x-2)

$\frac{x + 1}{x + 6} = \frac{x^{2} - 38 x}{x^{2} + 4 x - 12} - \frac{x - 6}{x - 2}$

Step 1

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

Subtract $\frac{x^{2} - 38 x}{x^{2} + 4 x - 12}$ from both sides of the equation.

$\frac{x + 1}{x + 6} - \frac{x^{2} - 38 x}{x^{2} + 4 x - 12} = - \frac{x - 6}{x - 2}$

Add $\frac{x - 6}{x - 2}$ to both sides of the equation.

$\frac{x + 1}{x + 6} - \frac{x^{2} - 38 x}{x^{2} + 4 x - 12} + \frac{x - 6}{x - 2} = 0$

Step 2

Simplify $\frac{x + 1}{x + 6} - \frac{x^{2} - 38 x}{x^{2} + 4 x - 12} + \frac{x - 6}{x - 2}$ .

Simplify each term.

Factor $x$ out of $x^{2} - 38 x$ .

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

$\frac{x + 1}{x + 6} - \frac{x \cdot x - 38 x}{x^{2} + 4 x - 12} + \frac{x - 6}{x - 2} = 0$

Factor $x$ out of $- 38 x$ .

$\frac{x + 1}{x + 6} - \frac{x \cdot x + x \cdot - 38}{x^{2} + 4 x - 12} + \frac{x - 6}{x - 2} = 0$

Factor $x$ out of $x \cdot x + x \cdot - 38$ .

$\frac{x + 1}{x + 6} - \frac{x (x - 38)}{x^{2} + 4 x - 12} + \frac{x - 6}{x - 2} = 0$

Factor $x^{2} + 4 x - 12$ using the AC method.

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 $4$ .

$- 2, 6$

Write the factored form using these integers.

$\frac{x + 1}{x + 6} - \frac{x (x - 38)}{(x - 2) (x + 6)} + \frac{x - 6}{x - 2} = 0$

To write $\frac{x + 1}{x + 6}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{x + 1}{x + 6} \cdot \frac{x - 2}{x - 2} - \frac{x (x - 38)}{(x - 2) (x + 6)} + \frac{x - 6}{x - 2} = 0$

Write each expression with a common denominator of $(x + 6) (x - 2)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x + 1}{x + 6}$ by $\frac{x - 2}{x - 2}$ .

$\frac{(x + 1) (x - 2)}{(x + 6) (x - 2)} - \frac{x (x - 38)}{(x - 2) (x + 6)} + \frac{x - 6}{x - 2} = 0$

Reorder the factors of $(x - 2) (x + 6)$ .

$\frac{(x + 1) (x - 2)}{(x + 6) (x - 2)} - \frac{x (x - 38)}{(x + 6) (x - 2)} + \frac{x - 6}{x - 2} = 0$

Combine the numerators over the common denominator.

$\frac{(x + 1) (x - 2) - x (x - 38)}{(x + 6) (x - 2)} + \frac{x - 6}{x - 2} = 0$

Simplify the numerator.

Expand $(x + 1) (x - 2)$ using the FOIL Method.

Apply the distributive property.

$\frac{x (x - 2) + 1 (x - 2) - x (x - 38)}{(x + 6) (x - 2)} + \frac{x - 6}{x - 2} = 0$

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 2 + 1 (x - 2) - x (x - 38)}{(x + 6) (x - 2)} + \frac{x - 6}{x - 2} = 0$

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 2 + 1 x + 1 \cdot - 2 - x (x - 38)}{(x + 6) (x - 2)} + \frac{x - 6}{x - 2} = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot - 2 + 1 x + 1 \cdot - 2 - x (x - 38)}{(x + 6) (x - 2)} + \frac{x - 6}{x - 2} = 0$

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

$\frac{x^{2} - 2 \cdot x + 1 x + 1 \cdot - 2 - x (x - 38)}{(x + 6) (x - 2)} + \frac{x - 6}{x - 2} = 0$

Multiply $x$ by $1$ .

$\frac{x^{2} - 2 x + x + 1 \cdot - 2 - x (x - 38)}{(x + 6) (x - 2)} + \frac{x - 6}{x - 2} = 0$

Multiply $- 2$ by $1$ .

$\frac{x^{2} - 2 x + x - 2 - x (x - 38)}{(x + 6) (x - 2)} + \frac{x - 6}{x - 2} = 0$

Add $- 2 x$ and $x$ .

$\frac{x^{2} - x - 2 - x (x - 38)}{(x + 6) (x - 2)} + \frac{x - 6}{x - 2} = 0$

Apply the distributive property.

$\frac{x^{2} - x - 2 - x \cdot x - x \cdot - 38}{(x + 6) (x - 2)} + \frac{x - 6}{x - 2} = 0$

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

Move $x$ .

$\frac{x^{2} - x - 2 - (x \cdot x) - x \cdot - 38}{(x + 6) (x - 2)} + \frac{x - 6}{x - 2} = 0$

Multiply $x$ by $x$ .

$\frac{x^{2} - x - 2 - x^{2} - x \cdot - 38}{(x + 6) (x - 2)} + \frac{x - 6}{x - 2} = 0$

Multiply $- 38$ by $- 1$ .

$\frac{x^{2} - x - 2 - x^{2} + 38 x}{(x + 6) (x - 2)} + \frac{x - 6}{x - 2} = 0$

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

$\frac{- x - 2 + 0 + 38 x}{(x + 6) (x - 2)} + \frac{x - 6}{x - 2} = 0$

Add $- x$ and $0$ .

$\frac{- x - 2 + 38 x}{(x + 6) (x - 2)} + \frac{x - 6}{x - 2} = 0$

Add $- x$ and $38 x$ .

$\frac{37 x - 2}{(x + 6) (x - 2)} + \frac{x - 6}{x - 2} = 0$

To write $\frac{x - 6}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x + 6}{x + 6}$ .

$\frac{37 x - 2}{(x + 6) (x - 2)} + \frac{x - 6}{x - 2} \cdot \frac{x + 6}{x + 6} = 0$

Write each expression with a common denominator of $(x + 6) (x - 2)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x - 6}{x - 2}$ by $\frac{x + 6}{x + 6}$ .

$\frac{37 x - 2}{(x + 6) (x - 2)} + \frac{(x - 6) (x + 6)}{(x - 2) (x + 6)} = 0$

Reorder the factors of $(x - 2) (x + 6)$ .

$\frac{37 x - 2}{(x + 6) (x - 2)} + \frac{(x - 6) (x + 6)}{(x + 6) (x - 2)} = 0$

Combine the numerators over the common denominator.

$\frac{37 x - 2 + (x - 6) (x + 6)}{(x + 6) (x - 2)} = 0$

Simplify the numerator.

Expand $(x - 6) (x + 6)$ using the FOIL Method.

Apply the distributive property.

$\frac{37 x - 2 + x (x + 6) - 6 (x + 6)}{(x + 6) (x - 2)} = 0$

Apply the distributive property.

$\frac{37 x - 2 + x \cdot x + x \cdot 6 - 6 (x + 6)}{(x + 6) (x - 2)} = 0$

Apply the distributive property.

$\frac{37 x - 2 + x \cdot x + x \cdot 6 - 6 x - 6 \cdot 6}{(x + 6) (x - 2)} = 0$

Combine the opposite terms in $x \cdot x + x \cdot 6 - 6 x - 6 \cdot 6$ .

Reorder the factors in the terms $x \cdot 6$ and $- 6 x$ .

$\frac{37 x - 2 + x \cdot x + 6 x - 6 x - 6 \cdot 6}{(x + 6) (x - 2)} = 0$

Subtract $6 x$ from $6 x$ .

$\frac{37 x - 2 + x \cdot x + 0 - 6 \cdot 6}{(x + 6) (x - 2)} = 0$

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

$\frac{37 x - 2 + x \cdot x - 6 \cdot 6}{(x + 6) (x - 2)} = 0$

Simplify each term.

Multiply $x$ by $x$ .

$\frac{37 x - 2 + x^{2} - 6 \cdot 6}{(x + 6) (x - 2)} = 0$

Multiply $- 6$ by $6$ .

$\frac{37 x - 2 + x^{2} - 36}{(x + 6) (x - 2)} = 0$

Subtract $36$ from $- 2$ .

$\frac{37 x + x^{2} - 38}{(x + 6) (x - 2)} = 0$

Reorder terms.

$\frac{x^{2} + 37 x - 38}{(x + 6) (x - 2)} = 0$

Factor $x^{2} + 37 x - 38$ using the AC method.

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 $- 38$ and whose sum is $37$ .

$- 1, 38$

Write the factored form using these integers.

$\frac{(x - 1) (x + 38)}{(x + 6) (x - 2)} = 0$

Step 3

Set the numerator equal to zero.

$(x - 1) (x + 38) = 0$

Step 4

Solve the equation for $x$ .

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

$x - 1 = 0$

$x + 38 = 0$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Set $x + 38$ equal to $0$ and solve for $x$ .

Set $x + 38$ equal to $0$ .

$x + 38 = 0$

Subtract $38$ from both sides of the equation.

$x = - 38$

The final solution is all the values that make $(x - 1) (x + 38) = 0$ true.

$x = 1, - 38$

Solve by Factoring (x+1)/(x+6)=(x^2-38x)/(x^2+4x-12)-(x-6)/(x-2)

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