Solve by Factoring (6x)/(x-6)-4/x=24/(x^2-6x)

$\frac{6 x}{x - 6} - \frac{4}{x} = \frac{24}{x^{2} - 6 x}$

Step 1

Subtract $\frac{24}{x^{2} - 6 x}$ from both sides of the equation.

$\frac{6 x}{x - 6} - \frac{4}{x} - \frac{24}{x^{2} - 6 x} = 0$

Step 2

Simplify $\frac{6 x}{x - 6} - \frac{4}{x} - \frac{24}{x^{2} - 6 x}$ .

Factor $x$ out of $x^{2} - 6 x$ .

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

$\frac{6 x}{x - 6} - \frac{4}{x} - \frac{24}{x \cdot x - 6 x} = 0$

Factor $x$ out of $- 6 x$ .

$\frac{6 x}{x - 6} - \frac{4}{x} - \frac{24}{x \cdot x + x \cdot - 6} = 0$

Factor $x$ out of $x \cdot x + x \cdot - 6$ .

$\frac{6 x}{x - 6} - \frac{4}{x} - \frac{24}{x (x - 6)} = 0$

To write $\frac{6 x}{x - 6}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{6 x}{x - 6} \cdot \frac{x}{x} - \frac{4}{x} - \frac{24}{x (x - 6)} = 0$

To write $- \frac{4}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 6}{x - 6}$ .

$\frac{6 x}{x - 6} \cdot \frac{x}{x} - \frac{4}{x} \cdot \frac{x - 6}{x - 6} - \frac{24}{x (x - 6)} = 0$

Write each expression with a common denominator of $(x - 6) x$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{6 x}{x - 6}$ by $\frac{x}{x}$ .

$\frac{6 x \cdot x}{(x - 6) x} - \frac{4}{x} \cdot \frac{x - 6}{x - 6} - \frac{24}{x (x - 6)} = 0$

Multiply $\frac{4}{x}$ by $\frac{x - 6}{x - 6}$ .

$\frac{6 x \cdot x}{(x - 6) x} - \frac{4 (x - 6)}{x (x - 6)} - \frac{24}{x (x - 6)} = 0$

Reorder the factors of $(x - 6) x$ .

$\frac{6 x \cdot x}{x (x - 6)} - \frac{4 (x - 6)}{x (x - 6)} - \frac{24}{x (x - 6)} = 0$

Combine the numerators over the common denominator.

$\frac{6 x \cdot x - 4 (x - 6)}{x (x - 6)} - \frac{24}{x (x - 6)} = 0$

Combine the numerators over the common denominator.

$\frac{6 x \cdot x - 4 (x - 6) - 24}{x (x - 6)} = 0$

Simplify each term.

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

Move $x$ .

$\frac{6 (x \cdot x) - 4 (x - 6) - 24}{x (x - 6)} = 0$

Multiply $x$ by $x$ .

$\frac{6 x^{2} - 4 (x - 6) - 24}{x (x - 6)} = 0$

Apply the distributive property.

$\frac{6 x^{2} - 4 x - 4 \cdot - 6 - 24}{x (x - 6)} = 0$

Multiply $- 4$ by $- 6$ .

$\frac{6 x^{2} - 4 x + 24 - 24}{x (x - 6)} = 0$

Combine the opposite terms in $6 x^{2} - 4 x + 24 - 24$ .

Subtract $24$ from $24$ .

$\frac{6 x^{2} - 4 x + 0}{x (x - 6)} = 0$

Add $6 x^{2} - 4 x$ and $0$ .

$\frac{6 x^{2} - 4 x}{x (x - 6)} = 0$

Factor $2 x$ out of $6 x^{2} - 4 x$ .

Factor $2 x$ out of $6 x^{2}$ .

$\frac{2 x (3 x) - 4 x}{x (x - 6)} = 0$

Factor $2 x$ out of $- 4 x$ .

$\frac{2 x (3 x) + 2 x (- 2)}{x (x - 6)} = 0$

Factor $2 x$ out of $2 x (3 x) + 2 x (- 2)$ .

$\frac{2 x (3 x - 2)}{x (x - 6)} = 0$

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{2 x (3 x - 2)}{x (x - 6)} = 0$

Rewrite the expression.

$\frac{2 (3 x - 2)}{x - 6} = 0$

Step 3

Set the numerator equal to zero.

$2 (3 x - 2) = 0$

Step 4

Solve the equation for $x$ .

Divide each term in $2 (3 x - 2) = 0$ by $2$ and simplify.

Divide each term in $2 (3 x - 2) = 0$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $3 x - 2$ by $1$ .

$3 x - 2 = \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$3 x - 2 = 0$

Add $2$ to both sides of the equation.

$3 x = 2$

Divide each term in $3 x = 2$ by $3$ and simplify.

Divide each term in $3 x = 2$ by $3$ .

$\frac{3 x}{3} = \frac{2}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{2}{3}$

Divide $x$ by $1$ .

$x = \frac{2}{3}$

Solve by Factoring (6x)/(x-6)-4/x=24/(x^2-6x)

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