Solve by Factoring (x+3)/(x+5)=(x^2-37x)/(x^2+x-20)-(x-5)/(x-4)

$\frac{x + 3}{x + 5} = \frac{x^{2} - 37 x}{x^{2} + x - 20} - \frac{x - 5}{x - 4}$

Step 1

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

Subtract $\frac{x^{2} - 37 x}{x^{2} + x - 20}$ from both sides of the equation.

$\frac{x + 3}{x + 5} - \frac{x^{2} - 37 x}{x^{2} + x - 20} = - \frac{x - 5}{x - 4}$

Add $\frac{x - 5}{x - 4}$ to both sides of the equation.

$\frac{x + 3}{x + 5} - \frac{x^{2} - 37 x}{x^{2} + x - 20} + \frac{x - 5}{x - 4} = 0$

Step 2

Simplify $\frac{x + 3}{x + 5} - \frac{x^{2} - 37 x}{x^{2} + x - 20} + \frac{x - 5}{x - 4}$ .

Simplify each term.

Factor $x$ out of $x^{2} - 37 x$ .

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

$\frac{x + 3}{x + 5} - \frac{x \cdot x - 37 x}{x^{2} + x - 20} + \frac{x - 5}{x - 4} = 0$

Factor $x$ out of $- 37 x$ .

$\frac{x + 3}{x + 5} - \frac{x \cdot x + x \cdot - 37}{x^{2} + x - 20} + \frac{x - 5}{x - 4} = 0$

Factor $x$ out of $x \cdot x + x \cdot - 37$ .

$\frac{x + 3}{x + 5} - \frac{x (x - 37)}{x^{2} + x - 20} + \frac{x - 5}{x - 4} = 0$

Factor $x^{2} + x - 20$ 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 $- 20$ and whose sum is $1$ .

$- 4, 5$

Write the factored form using these integers.

$\frac{x + 3}{x + 5} - \frac{x (x - 37)}{(x - 4) (x + 5)} + \frac{x - 5}{x - 4} = 0$

To write $\frac{x + 3}{x + 5}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{x + 3}{x + 5} \cdot \frac{x - 4}{x - 4} - \frac{x (x - 37)}{(x - 4) (x + 5)} + \frac{x - 5}{x - 4} = 0$

Write each expression with a common denominator of $(x + 5) (x - 4)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x + 3}{x + 5}$ by $\frac{x - 4}{x - 4}$ .

$\frac{(x + 3) (x - 4)}{(x + 5) (x - 4)} - \frac{x (x - 37)}{(x - 4) (x + 5)} + \frac{x - 5}{x - 4} = 0$

Reorder the factors of $(x - 4) (x + 5)$ .

$\frac{(x + 3) (x - 4)}{(x + 5) (x - 4)} - \frac{x (x - 37)}{(x + 5) (x - 4)} + \frac{x - 5}{x - 4} = 0$

Combine the numerators over the common denominator.

$\frac{(x + 3) (x - 4) - x (x - 37)}{(x + 5) (x - 4)} + \frac{x - 5}{x - 4} = 0$

Simplify the numerator.

Expand $(x + 3) (x - 4)$ using the FOIL Method.

Apply the distributive property.

$\frac{x (x - 4) + 3 (x - 4) - x (x - 37)}{(x + 5) (x - 4)} + \frac{x - 5}{x - 4} = 0$

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 4 + 3 (x - 4) - x (x - 37)}{(x + 5) (x - 4)} + \frac{x - 5}{x - 4} = 0$

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 4 + 3 x + 3 \cdot - 4 - x (x - 37)}{(x + 5) (x - 4)} + \frac{x - 5}{x - 4} = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot - 4 + 3 x + 3 \cdot - 4 - x (x - 37)}{(x + 5) (x - 4)} + \frac{x - 5}{x - 4} = 0$

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

$\frac{x^{2} - 4 \cdot x + 3 x + 3 \cdot - 4 - x (x - 37)}{(x + 5) (x - 4)} + \frac{x - 5}{x - 4} = 0$

Multiply $3$ by $- 4$ .

$\frac{x^{2} - 4 x + 3 x - 12 - x (x - 37)}{(x + 5) (x - 4)} + \frac{x - 5}{x - 4} = 0$

Add $- 4 x$ and $3 x$ .

$\frac{x^{2} - x - 12 - x (x - 37)}{(x + 5) (x - 4)} + \frac{x - 5}{x - 4} = 0$

Apply the distributive property.

$\frac{x^{2} - x - 12 - x \cdot x - x \cdot - 37}{(x + 5) (x - 4)} + \frac{x - 5}{x - 4} = 0$

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

Move $x$ .

$\frac{x^{2} - x - 12 - (x \cdot x) - x \cdot - 37}{(x + 5) (x - 4)} + \frac{x - 5}{x - 4} = 0$

Multiply $x$ by $x$ .

$\frac{x^{2} - x - 12 - x^{2} - x \cdot - 37}{(x + 5) (x - 4)} + \frac{x - 5}{x - 4} = 0$

Multiply $- 37$ by $- 1$ .

$\frac{x^{2} - x - 12 - x^{2} + 37 x}{(x + 5) (x - 4)} + \frac{x - 5}{x - 4} = 0$

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

$\frac{- x - 12 + 0 + 37 x}{(x + 5) (x - 4)} + \frac{x - 5}{x - 4} = 0$

Add $- x$ and $0$ .

$\frac{- x - 12 + 37 x}{(x + 5) (x - 4)} + \frac{x - 5}{x - 4} = 0$

Add $- x$ and $37 x$ .

$\frac{36 x - 12}{(x + 5) (x - 4)} + \frac{x - 5}{x - 4} = 0$

Factor $12$ out of $36 x - 12$ .

Factor $12$ out of $36 x$ .

$\frac{12 (3 x) - 12}{(x + 5) (x - 4)} + \frac{x - 5}{x - 4} = 0$

Factor $12$ out of $- 12$ .

$\frac{12 (3 x) + 12 (- 1)}{(x + 5) (x - 4)} + \frac{x - 5}{x - 4} = 0$

Factor $12$ out of $12 (3 x) + 12 (- 1)$ .

$\frac{12 (3 x - 1)}{(x + 5) (x - 4)} + \frac{x - 5}{x - 4} = 0$

To write $\frac{x - 5}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$\frac{12 (3 x - 1)}{(x + 5) (x - 4)} + \frac{x - 5}{x - 4} \cdot \frac{x + 5}{x + 5} = 0$

Write each expression with a common denominator of $(x + 5) (x - 4)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x - 5}{x - 4}$ by $\frac{x + 5}{x + 5}$ .

$\frac{12 (3 x - 1)}{(x + 5) (x - 4)} + \frac{(x - 5) (x + 5)}{(x - 4) (x + 5)} = 0$

Reorder the factors of $(x - 4) (x + 5)$ .

$\frac{12 (3 x - 1)}{(x + 5) (x - 4)} + \frac{(x - 5) (x + 5)}{(x + 5) (x - 4)} = 0$

Combine the numerators over the common denominator.

$\frac{12 (3 x - 1) + (x - 5) (x + 5)}{(x + 5) (x - 4)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{12 (3 x) + 12 \cdot - 1 + (x - 5) (x + 5)}{(x + 5) (x - 4)} = 0$

Multiply $3$ by $12$ .

$\frac{36 x + 12 \cdot - 1 + (x - 5) (x + 5)}{(x + 5) (x - 4)} = 0$

Multiply $12$ by $- 1$ .

$\frac{36 x - 12 + (x - 5) (x + 5)}{(x + 5) (x - 4)} = 0$

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

Apply the distributive property.

$\frac{36 x - 12 + x (x + 5) - 5 (x + 5)}{(x + 5) (x - 4)} = 0$

Apply the distributive property.

$\frac{36 x - 12 + x \cdot x + x \cdot 5 - 5 (x + 5)}{(x + 5) (x - 4)} = 0$

Apply the distributive property.

$\frac{36 x - 12 + x \cdot x + x \cdot 5 - 5 x - 5 \cdot 5}{(x + 5) (x - 4)} = 0$

Combine the opposite terms in $x \cdot x + x \cdot 5 - 5 x - 5 \cdot 5$ .

Reorder the factors in the terms $x \cdot 5$ and $- 5 x$ .

$\frac{36 x - 12 + x \cdot x + 5 x - 5 x - 5 \cdot 5}{(x + 5) (x - 4)} = 0$

Subtract $5 x$ from $5 x$ .

$\frac{36 x - 12 + x \cdot x + 0 - 5 \cdot 5}{(x + 5) (x - 4)} = 0$

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

$\frac{36 x - 12 + x \cdot x - 5 \cdot 5}{(x + 5) (x - 4)} = 0$

Simplify each term.

Multiply $x$ by $x$ .

$\frac{36 x - 12 + x^{2} - 5 \cdot 5}{(x + 5) (x - 4)} = 0$

Multiply $- 5$ by $5$ .

$\frac{36 x - 12 + x^{2} - 25}{(x + 5) (x - 4)} = 0$

Subtract $25$ from $- 12$ .

$\frac{36 x + x^{2} - 37}{(x + 5) (x - 4)} = 0$

Reorder terms.

$\frac{x^{2} + 36 x - 37}{(x + 5) (x - 4)} = 0$

Factor $x^{2} + 36 x - 37$ 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 $- 37$ and whose sum is $36$ .

$- 1, 37$

Write the factored form using these integers.

$\frac{(x - 1) (x + 37)}{(x + 5) (x - 4)} = 0$

Step 3

Set the numerator equal to zero.

$(x - 1) (x + 37) = 0$

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 $0$ .

$x - 1 = 0$

$x + 37 = 0$

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 $x + 37$ equal to $0$ and solve for $x$ .

Set $x + 37$ equal to $0$ .

$x + 37 = 0$

Subtract $37$ from both sides of the equation.

$x = - 37$

The final solution is all the values that make $(x - 1) (x + 37) = 0$ true.

$x = 1, - 37$

Solve by Factoring (x+3)/(x+5)=(x^2-37x)/(x^2+x-20)-(x-5)/(x-4)

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