Solve by Factoring 15/x-15/(x-2)=-2

$\frac{15}{x} - \frac{15}{x - 2} = - 2$

Step 1

Add $2$ to both sides of the equation.

$\frac{15}{x} - \frac{15}{x - 2} + 2 = 0$

Step 2

Simplify $\frac{15}{x} - \frac{15}{x - 2} + 2$ .

To write $\frac{15}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{15}{x} \cdot \frac{x - 2}{x - 2} - \frac{15}{x - 2} + 2 = 0$

To write $- \frac{15}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{15}{x} \cdot \frac{x - 2}{x - 2} - \frac{15}{x - 2} \cdot \frac{x}{x} + 2 = 0$

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

Multiply $\frac{15}{x}$ by $\frac{x - 2}{x - 2}$ .

$\frac{15 (x - 2)}{x (x - 2)} - \frac{15}{x - 2} \cdot \frac{x}{x} + 2 = 0$

Multiply $\frac{15}{x - 2}$ by $\frac{x}{x}$ .

$\frac{15 (x - 2)}{x (x - 2)} - \frac{15 x}{(x - 2) x} + 2 = 0$

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

$\frac{15 (x - 2)}{x (x - 2)} - \frac{15 x}{x (x - 2)} + 2 = 0$

Combine the numerators over the common denominator.

$\frac{15 (x - 2) - 15 x}{x (x - 2)} + 2 = 0$

Simplify each term.

Simplify the numerator.

Factor $15$ out of $15 (x - 2) - 15 x$ .

Factor $15$ out of $- 15 x$ .

$\frac{15 (x - 2) + 15 (- x)}{x (x - 2)} + 2 = 0$

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

$\frac{15 (x - 2 - x)}{x (x - 2)} + 2 = 0$

Subtract $x$ from $x$ .

$\frac{15 (0 - 2)}{x (x - 2)} + 2 = 0$

Subtract $2$ from $0$ .

$\frac{15 \cdot - 2}{x (x - 2)} + 2 = 0$

Multiply $15$ by $- 2$ .

$\frac{- 30}{x (x - 2)} + 2 = 0$

Move the negative in front of the fraction.

$- \frac{30}{x (x - 2)} + 2 = 0$

Step 3

Factor $2$ out of $- \frac{30}{x (x - 2)} + 2$ .

Factor $2$ out of $- \frac{30}{x (x - 2)}$ .

$2 (- \frac{15}{x (x - 2)}) + 2 = 0$

Factor $2$ out of $2$ .

$2 (- \frac{15}{x (x - 2)}) + 2 (1) = 0$

Factor $2$ out of $2 (- \frac{15}{x (x - 2)}) + 2 (1)$ .

$2 (- \frac{15}{x (x - 2)} + 1) = 0$

Step 4

Write $1$ as a fraction with a common denominator.

$2 (- \frac{15}{x (x - 2)} + \frac{x (x - 2)}{x (x - 2)}) = 0$

Step 5

Combine the numerators over the common denominator.

$2 (\frac{- 15 + x (x - 2)}{x (x - 2)}) = 0$

Step 6

Simplify the numerator.

Apply the distributive property.

$2 (\frac{- 15 + x \cdot x + x \cdot - 2}{x (x - 2)}) = 0$

Multiply $x$ by $x$ .

$2 (\frac{- 15 + x^{2} + x \cdot - 2}{x (x - 2)}) = 0$

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

$2 (\frac{- 15 + x^{2} - 2 x}{x (x - 2)}) = 0$

Reorder terms.

$2 (\frac{x^{2} - 2 x - 15}{x (x - 2)}) = 0$

Factor $x^{2} - 2 x - 15$ 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 $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Write the factored form using these integers.

$2 (\frac{(x - 5) (x + 3)}{x (x - 2)}) = 0$

Step 7

Combine $2$ and $\frac{(x - 5) (x + 3)}{x (x - 2)}$ .

$\frac{2 (x - 5) (x + 3)}{x (x - 2)} = 0$

Step 8

Set the numerator equal to zero.

$2 (x - 5) (x + 3) = 0$

Step 9

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 $0$ .

$x - 5 = 0$

$x + 3 = 0$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Set $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

The final solution is all the values that make $2 (x - 5) (x + 3) = 0$ true.

$x = 5, - 3$

Solve by Factoring 15/x-15/(x-2)=-2

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