Solve by Factoring (7(x-2))/(x-3)+5/x=-5/(x(x-3))

$\frac{7 (x - 2)}{x - 3} + \frac{5}{x} = \frac{- 5}{x (x - 3)}$

Step 1

Subtract $\frac{- 5}{x (x - 3)}$ from both sides of the equation.

$\frac{7 (x - 2)}{x - 3} + \frac{5}{x} - \frac{- 5}{x (x - 3)} = 0$

Step 2

Simplify $\frac{7 (x - 2)}{x - 3} + \frac{5}{x} - \frac{- 5}{x (x - 3)}$ .

Simplify each term.

Move the negative in front of the fraction.

$\frac{7 (x - 2)}{x - 3} + \frac{5}{x} - - \frac{5}{x (x - 3)} = 0$

Multiply $- - \frac{5}{x (x - 3)}$ .

Multiply $- 1$ by $- 1$ .

$\frac{7 (x - 2)}{x - 3} + \frac{5}{x} + 1 \frac{5}{x (x - 3)} = 0$

Multiply $\frac{5}{x (x - 3)}$ by $1$ .

$\frac{7 (x - 2)}{x - 3} + \frac{5}{x} + \frac{5}{x (x - 3)} = 0$

To write $\frac{7 (x - 2)}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{7 (x - 2)}{x - 3} \cdot \frac{x}{x} + \frac{5}{x} + \frac{5}{x (x - 3)} = 0$

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

$\frac{7 (x - 2)}{x - 3} \cdot \frac{x}{x} + \frac{5}{x} \cdot \frac{x - 3}{x - 3} + \frac{5}{x (x - 3)} = 0$

Write each expression with a common denominator of $(x - 3) x$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{7 (x - 2)}{x - 3}$ by $\frac{x}{x}$ .

$\frac{7 (x - 2) x}{(x - 3) x} + \frac{5}{x} \cdot \frac{x - 3}{x - 3} + \frac{5}{x (x - 3)} = 0$

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

$\frac{7 (x - 2) x}{(x - 3) x} + \frac{5 (x - 3)}{x (x - 3)} + \frac{5}{x (x - 3)} = 0$

Reorder the factors of $(x - 3) x$ .

$\frac{7 (x - 2) x}{x (x - 3)} + \frac{5 (x - 3)}{x (x - 3)} + \frac{5}{x (x - 3)} = 0$

Combine the numerators over the common denominator.

$\frac{7 (x - 2) x + 5 (x - 3)}{x (x - 3)} + \frac{5}{x (x - 3)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{(7 x + 7 \cdot - 2) x + 5 (x - 3)}{x (x - 3)} + \frac{5}{x (x - 3)} = 0$

Multiply $7$ by $- 2$ .

$\frac{(7 x - 14) x + 5 (x - 3)}{x (x - 3)} + \frac{5}{x (x - 3)} = 0$

Apply the distributive property.

$\frac{7 x \cdot x - 14 x + 5 (x - 3)}{x (x - 3)} + \frac{5}{x (x - 3)} = 0$

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

Move $x$ .

$\frac{7 (x \cdot x) - 14 x + 5 (x - 3)}{x (x - 3)} + \frac{5}{x (x - 3)} = 0$

Multiply $x$ by $x$ .

$\frac{7 x^{2} - 14 x + 5 (x - 3)}{x (x - 3)} + \frac{5}{x (x - 3)} = 0$

Apply the distributive property.

$\frac{7 x^{2} - 14 x + 5 x + 5 \cdot - 3}{x (x - 3)} + \frac{5}{x (x - 3)} = 0$

Multiply $5$ by $- 3$ .

$\frac{7 x^{2} - 14 x + 5 x - 15}{x (x - 3)} + \frac{5}{x (x - 3)} = 0$

Add $- 14 x$ and $5 x$ .

$\frac{7 x^{2} - 9 x - 15}{x (x - 3)} + \frac{5}{x (x - 3)} = 0$

Combine the numerators over the common denominator.

$\frac{7 x^{2} - 9 x - 15 + 5}{x (x - 3)} = 0$

Add $- 15$ and $5$ .

$\frac{7 x^{2} - 9 x - 10}{x (x - 3)} = 0$

Factor by grouping.

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 $a \cdot c = 7 \cdot - 10 = - 70$ and whose sum is $b = - 9$ .

Factor $- 9$ out of $- 9 x$ .

$\frac{7 x^{2} - 9 (x) - 10}{x (x - 3)} = 0$

Rewrite $- 9$ as $5$ plus $- 14$

$\frac{7 x^{2} + (5 - 14) x - 10}{x (x - 3)} = 0$

Apply the distributive property.

$\frac{7 x^{2} + 5 x - 14 x - 10}{x (x - 3)} = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(7 x^{2} + 5 x) - 14 x - 10}{x (x - 3)} = 0$

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

$\frac{x (7 x + 5) - 2 (7 x + 5)}{x (x - 3)} = 0$

Factor the polynomial by factoring out the greatest common factor, $7 x + 5$ .

$\frac{(7 x + 5) (x - 2)}{x (x - 3)} = 0$

Step 3

Set the numerator equal to zero.

$(7 x + 5) (x - 2) = 0$

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 $0$ .

$7 x + 5 = 0$

$x - 2 = 0$

Set $7 x + 5$ equal to $0$ and solve for $x$ .

Set $7 x + 5$ equal to $0$ .

$7 x + 5 = 0$

Solve $7 x + 5 = 0$ for $x$ .

Subtract $5$ from both sides of the equation.

$7 x = - 5$

Divide each term in $7 x = - 5$ by $7$ and simplify.

Divide each term in $7 x = - 5$ by $7$ .

$\frac{7 x}{7} = \frac{- 5}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x}{7} = \frac{- 5}{7}$

Divide $x$ by $1$ .

$x = \frac{- 5}{7}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{5}{7}$

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

The final solution is all the values that make $(7 x + 5) (x - 2) = 0$ true.

$x = - \frac{5}{7}, 2$

Solve by Factoring (7(x-2))/(x-3)+5/x=-5/(x(x-3))

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