Solve by Factoring (x+1)/(x+7)=(x^2-51x)/(x^2+5x-14)-(x-7)/(x-2)

$\frac{x + 1}{x + 7} = \frac{x^{2} - 51 x}{x^{2} + 5 x - 14} - \frac{x - 7}{x - 2}$

Step 1

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

Subtract $\frac{x^{2} - 51 x}{x^{2} + 5 x - 14}$ from both sides of the equation.

$\frac{x + 1}{x + 7} - \frac{x^{2} - 51 x}{x^{2} + 5 x - 14} = - \frac{x - 7}{x - 2}$

Add $\frac{x - 7}{x - 2}$ to both sides of the equation.

$\frac{x + 1}{x + 7} - \frac{x^{2} - 51 x}{x^{2} + 5 x - 14} + \frac{x - 7}{x - 2} = 0$

Step 2

Simplify $\frac{x + 1}{x + 7} - \frac{x^{2} - 51 x}{x^{2} + 5 x - 14} + \frac{x - 7}{x - 2}$ .

Simplify each term.

Factor $x$ out of $x^{2} - 51 x$ .

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

$\frac{x + 1}{x + 7} - \frac{x \cdot x - 51 x}{x^{2} + 5 x - 14} + \frac{x - 7}{x - 2} = 0$

Factor $x$ out of $- 51 x$ .

$\frac{x + 1}{x + 7} - \frac{x \cdot x + x \cdot - 51}{x^{2} + 5 x - 14} + \frac{x - 7}{x - 2} = 0$

Factor $x$ out of $x \cdot x + x \cdot - 51$ .

$\frac{x + 1}{x + 7} - \frac{x (x - 51)}{x^{2} + 5 x - 14} + \frac{x - 7}{x - 2} = 0$

Factor $x^{2} + 5 x - 14$ 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 $- 14$ and whose sum is $5$ .

$- 2, 7$

Write the factored form using these integers.

$\frac{x + 1}{x + 7} - \frac{x (x - 51)}{(x - 2) (x + 7)} + \frac{x - 7}{x - 2} = 0$

To write $\frac{x + 1}{x + 7}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{x + 1}{x + 7} \cdot \frac{x - 2}{x - 2} - \frac{x (x - 51)}{(x - 2) (x + 7)} + \frac{x - 7}{x - 2} = 0$

Write each expression with a common denominator of $(x + 7) (x - 2)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x + 1}{x + 7}$ by $\frac{x - 2}{x - 2}$ .

$\frac{(x + 1) (x - 2)}{(x + 7) (x - 2)} - \frac{x (x - 51)}{(x - 2) (x + 7)} + \frac{x - 7}{x - 2} = 0$

Reorder the factors of $(x - 2) (x + 7)$ .

$\frac{(x + 1) (x - 2)}{(x + 7) (x - 2)} - \frac{x (x - 51)}{(x + 7) (x - 2)} + \frac{x - 7}{x - 2} = 0$

Combine the numerators over the common denominator.

$\frac{(x + 1) (x - 2) - x (x - 51)}{(x + 7) (x - 2)} + \frac{x - 7}{x - 2} = 0$

Simplify the numerator.

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

Apply the distributive property.

$\frac{x (x - 2) + 1 (x - 2) - x (x - 51)}{(x + 7) (x - 2)} + \frac{x - 7}{x - 2} = 0$

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 2 + 1 (x - 2) - x (x - 51)}{(x + 7) (x - 2)} + \frac{x - 7}{x - 2} = 0$

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 2 + 1 x + 1 \cdot - 2 - x (x - 51)}{(x + 7) (x - 2)} + \frac{x - 7}{x - 2} = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot - 2 + 1 x + 1 \cdot - 2 - x (x - 51)}{(x + 7) (x - 2)} + \frac{x - 7}{x - 2} = 0$

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

$\frac{x^{2} - 2 \cdot x + 1 x + 1 \cdot - 2 - x (x - 51)}{(x + 7) (x - 2)} + \frac{x - 7}{x - 2} = 0$

Multiply $x$ by $1$ .

$\frac{x^{2} - 2 x + x + 1 \cdot - 2 - x (x - 51)}{(x + 7) (x - 2)} + \frac{x - 7}{x - 2} = 0$

Multiply $- 2$ by $1$ .

$\frac{x^{2} - 2 x + x - 2 - x (x - 51)}{(x + 7) (x - 2)} + \frac{x - 7}{x - 2} = 0$

Add $- 2 x$ and $x$ .

$\frac{x^{2} - x - 2 - x (x - 51)}{(x + 7) (x - 2)} + \frac{x - 7}{x - 2} = 0$

Apply the distributive property.

$\frac{x^{2} - x - 2 - x \cdot x - x \cdot - 51}{(x + 7) (x - 2)} + \frac{x - 7}{x - 2} = 0$

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

Move $x$ .

$\frac{x^{2} - x - 2 - (x \cdot x) - x \cdot - 51}{(x + 7) (x - 2)} + \frac{x - 7}{x - 2} = 0$

Multiply $x$ by $x$ .

$\frac{x^{2} - x - 2 - x^{2} - x \cdot - 51}{(x + 7) (x - 2)} + \frac{x - 7}{x - 2} = 0$

Multiply $- 51$ by $- 1$ .

$\frac{x^{2} - x - 2 - x^{2} + 51 x}{(x + 7) (x - 2)} + \frac{x - 7}{x - 2} = 0$

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

$\frac{- x - 2 + 0 + 51 x}{(x + 7) (x - 2)} + \frac{x - 7}{x - 2} = 0$

Add $- x$ and $0$ .

$\frac{- x - 2 + 51 x}{(x + 7) (x - 2)} + \frac{x - 7}{x - 2} = 0$

Add $- x$ and $51 x$ .

$\frac{50 x - 2}{(x + 7) (x - 2)} + \frac{x - 7}{x - 2} = 0$

Factor $2$ out of $50 x - 2$ .

Factor $2$ out of $50 x$ .

$\frac{2 (25 x) - 2}{(x + 7) (x - 2)} + \frac{x - 7}{x - 2} = 0$

Factor $2$ out of $- 2$ .

$\frac{2 (25 x) + 2 (- 1)}{(x + 7) (x - 2)} + \frac{x - 7}{x - 2} = 0$

Factor $2$ out of $2 (25 x) + 2 (- 1)$ .

$\frac{2 (25 x - 1)}{(x + 7) (x - 2)} + \frac{x - 7}{x - 2} = 0$

To write $\frac{x - 7}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x + 7}{x + 7}$ .

$\frac{2 (25 x - 1)}{(x + 7) (x - 2)} + \frac{x - 7}{x - 2} \cdot \frac{x + 7}{x + 7} = 0$

Write each expression with a common denominator of $(x + 7) (x - 2)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x - 7}{x - 2}$ by $\frac{x + 7}{x + 7}$ .

$\frac{2 (25 x - 1)}{(x + 7) (x - 2)} + \frac{(x - 7) (x + 7)}{(x - 2) (x + 7)} = 0$

Reorder the factors of $(x - 2) (x + 7)$ .

$\frac{2 (25 x - 1)}{(x + 7) (x - 2)} + \frac{(x - 7) (x + 7)}{(x + 7) (x - 2)} = 0$

Combine the numerators over the common denominator.

$\frac{2 (25 x - 1) + (x - 7) (x + 7)}{(x + 7) (x - 2)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{2 (25 x) + 2 \cdot - 1 + (x - 7) (x + 7)}{(x + 7) (x - 2)} = 0$

Multiply $25$ by $2$ .

$\frac{50 x + 2 \cdot - 1 + (x - 7) (x + 7)}{(x + 7) (x - 2)} = 0$

Multiply $2$ by $- 1$ .

$\frac{50 x - 2 + (x - 7) (x + 7)}{(x + 7) (x - 2)} = 0$

Expand $(x - 7) (x + 7)$ using the FOIL Method.

Apply the distributive property.

$\frac{50 x - 2 + x (x + 7) - 7 (x + 7)}{(x + 7) (x - 2)} = 0$

Apply the distributive property.

$\frac{50 x - 2 + x \cdot x + x \cdot 7 - 7 (x + 7)}{(x + 7) (x - 2)} = 0$

Apply the distributive property.

$\frac{50 x - 2 + x \cdot x + x \cdot 7 - 7 x - 7 \cdot 7}{(x + 7) (x - 2)} = 0$

Combine the opposite terms in $x \cdot x + x \cdot 7 - 7 x - 7 \cdot 7$ .

Reorder the factors in the terms $x \cdot 7$ and $- 7 x$ .

$\frac{50 x - 2 + x \cdot x + 7 x - 7 x - 7 \cdot 7}{(x + 7) (x - 2)} = 0$

Subtract $7 x$ from $7 x$ .

$\frac{50 x - 2 + x \cdot x + 0 - 7 \cdot 7}{(x + 7) (x - 2)} = 0$

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

$\frac{50 x - 2 + x \cdot x - 7 \cdot 7}{(x + 7) (x - 2)} = 0$

Simplify each term.

Multiply $x$ by $x$ .

$\frac{50 x - 2 + x^{2} - 7 \cdot 7}{(x + 7) (x - 2)} = 0$

Multiply $- 7$ by $7$ .

$\frac{50 x - 2 + x^{2} - 49}{(x + 7) (x - 2)} = 0$

Subtract $49$ from $- 2$ .

$\frac{50 x + x^{2} - 51}{(x + 7) (x - 2)} = 0$

Reorder terms.

$\frac{x^{2} + 50 x - 51}{(x + 7) (x - 2)} = 0$

Factor $x^{2} + 50 x - 51$ 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 $- 51$ and whose sum is $50$ .

$- 1, 51$

Write the factored form using these integers.

$\frac{(x - 1) (x + 51)}{(x + 7) (x - 2)} = 0$

Step 3

Set the numerator equal to zero.

$(x - 1) (x + 51) = 0$

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 $0$ .

$x - 1 = 0$

$x + 51 = 0$

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 $x + 51$ equal to $0$ and solve for $x$ .

Set $x + 51$ equal to $0$ .

$x + 51 = 0$

Subtract $51$ from both sides of the equation.

$x = - 51$

The final solution is all the values that make $(x - 1) (x + 51) = 0$ true.

$x = 1, - 51$

Solve by Factoring (x+1)/(x+7)=(x^2-51x)/(x^2+5x-14)-(x-7)/(x-2)

Related Questions

Support LCMGCF.com by disabling your adblocker.