Solve by Factoring 1/x+(3x+12)/(x^2-5x)=(7x-56)/(x^2-5x)

$\frac{1}{x} + \frac{3 x + 12}{x^{2} - 5 x} = \frac{7 x - 56}{x^{2} - 5 x}$

Step 1

Subtract $\frac{7 x - 56}{x^{2} - 5 x}$ from both sides of the equation.

$\frac{1}{x} + \frac{3 x + 12}{x^{2} - 5 x} - \frac{7 x - 56}{x^{2} - 5 x} = 0$

Step 2

Simplify $\frac{1}{x} + \frac{3 x + 12}{x^{2} - 5 x} - \frac{7 x - 56}{x^{2} - 5 x}$ .

Combine the numerators over the common denominator.

$\frac{1}{x} + \frac{3 x + 12 - (7 x - 56)}{x^{2} - 5 x} = 0$

Simplify each term.

Apply the distributive property.

$\frac{1}{x} + \frac{3 x + 12 - (7 x) - - 56}{x^{2} - 5 x} = 0$

Multiply $7$ by $- 1$ .

$\frac{1}{x} + \frac{3 x + 12 - 7 x - - 56}{x^{2} - 5 x} = 0$

Multiply $- 1$ by $- 56$ .

$\frac{1}{x} + \frac{3 x + 12 - 7 x + 56}{x^{2} - 5 x} = 0$

Subtract $7 x$ from $3 x$ .

$\frac{1}{x} + \frac{- 4 x + 12 + 56}{x^{2} - 5 x} = 0$

Add $12$ and $56$ .

$\frac{1}{x} + \frac{- 4 x + 68}{x^{2} - 5 x} = 0$

Simplify each term.

Factor $4$ out of $- 4 x + 68$ .

Factor $4$ out of $- 4 x$ .

$\frac{1}{x} + \frac{4 (- x) + 68}{x^{2} - 5 x} = 0$

Factor $4$ out of $68$ .

$\frac{1}{x} + \frac{4 (- x) + 4 (17)}{x^{2} - 5 x} = 0$

Factor $4$ out of $4 (- x) + 4 (17)$ .

$\frac{1}{x} + \frac{4 (- x + 17)}{x^{2} - 5 x} = 0$

Factor $x$ out of $x^{2} - 5 x$ .

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

$\frac{1}{x} + \frac{4 (- x + 17)}{x \cdot x - 5 x} = 0$

Factor $x$ out of $- 5 x$ .

$\frac{1}{x} + \frac{4 (- x + 17)}{x \cdot x + x \cdot - 5} = 0$

Factor $x$ out of $x \cdot x + x \cdot - 5$ .

$\frac{1}{x} + \frac{4 (- x + 17)}{x (x - 5)} = 0$

To write $\frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$\frac{1}{x} \cdot \frac{x - 5}{x - 5} + \frac{4 (- x + 17)}{x (x - 5)} = 0$

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

$\frac{x - 5}{x (x - 5)} + \frac{4 (- x + 17)}{x (x - 5)} = 0$

Combine the numerators over the common denominator.

$\frac{x - 5 + 4 (- x + 17)}{x (x - 5)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{x - 5 + 4 (- x) + 4 \cdot 17}{x (x - 5)} = 0$

Multiply $- 1$ by $4$ .

$\frac{x - 5 - 4 x + 4 \cdot 17}{x (x - 5)} = 0$

Multiply $4$ by $17$ .

$\frac{x - 5 - 4 x + 68}{x (x - 5)} = 0$

Subtract $4 x$ from $x$ .

$\frac{- 3 x - 5 + 68}{x (x - 5)} = 0$

Add $- 5$ and $68$ .

$\frac{- 3 x + 63}{x (x - 5)} = 0$

Factor $3$ out of $- 3 x + 63$ .

Factor $3$ out of $- 3 x$ .

$\frac{3 (- x) + 63}{x (x - 5)} = 0$

Factor $3$ out of $63$ .

$\frac{3 (- x) + 3 (21)}{x (x - 5)} = 0$

Factor $3$ out of $3 (- x) + 3 (21)$ .

$\frac{3 (- x + 21)}{x (x - 5)} = 0$

Factor $- 1$ out of $- x$ .

$\frac{3 (- (x) + 21)}{x (x - 5)} = 0$

Rewrite $21$ as $- 1 (- 21)$ .

$\frac{3 (- (x) - 1 (- 21))}{x (x - 5)} = 0$

Factor $- 1$ out of $- (x) - 1 (- 21)$ .

$\frac{3 (- (x - 21))}{x (x - 5)} = 0$

Rewrite $- (x - 21)$ as $- 1 (x - 21)$ .

$\frac{3 (- 1 (x - 21))}{x (x - 5)} = 0$

Move the negative in front of the fraction.

$- \frac{3 (x - 21)}{x (x - 5)} = 0$

Step 3

Set the numerator equal to zero.

$3 (x - 21) = 0$

Step 4

Solve the equation for $x$ .

Divide each term in $3 (x - 21) = 0$ by $3$ and simplify.

Divide each term in $3 (x - 21) = 0$ by $3$ .

$\frac{3 (x - 21)}{3} = \frac{0}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 (x - 21)}{3} = \frac{0}{3}$

Divide $x - 21$ by $1$ .

$x - 21 = \frac{0}{3}$

Simplify the right side.

Divide $0$ by $3$ .

$x - 21 = 0$

Add $21$ to both sides of the equation.

$x = 21$

Solve by Factoring 1/x+(3x+12)/(x^2-5x)=(7x-56)/(x^2-5x)

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