Algebra Examples

$\frac{3}{x^{2} - 3 x} + \frac{4}{x} = \frac{1}{x - 3}$

Step 1

Subtract $\frac{1}{x - 3}$ from both sides of the equation.

$\frac{3}{x^{2} - 3 x} + \frac{4}{x} - \frac{1}{x - 3} = 0$

Step 2

Simplify $\frac{3}{x^{2} - 3 x} + \frac{4}{x} - \frac{1}{x - 3}$ .

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Step 2.1

Factor $x$ out of $x^{2} - 3 x$ .

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Step 2.1.1

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

$\frac{3}{x \cdot x - 3 x} + \frac{4}{x} - \frac{1}{x - 3} = 0$

Step 2.1.2

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

$\frac{3}{x \cdot x + x \cdot - 3} + \frac{4}{x} - \frac{1}{x - 3} = 0$

Step 2.1.3

Factor $x$ out of $x \cdot x + x \cdot - 3$ .

$\frac{3}{x (x - 3)} + \frac{4}{x} - \frac{1}{x - 3} = 0$

Step 2.2

To write $- \frac{1}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{3}{x (x - 3)} - \frac{1}{x - 3} \cdot \frac{x}{x} + \frac{4}{x} = 0$

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify each term.

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Step 2.5.1

Cancel the common factor of $3 - x$ and $x - 3$ .

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Step 2.5.1.1

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{- 1 (- 3) - x}{x (x - 3)} + \frac{4}{x} = 0$

Step 2.5.1.2

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

$\frac{- 1 (- 3) - (x)}{x (x - 3)} + \frac{4}{x} = 0$

Step 2.5.1.3

Factor $- 1$ out of $- 1 (- 3) - (x)$ .

$\frac{- 1 (- 3 + x)}{x (x - 3)} + \frac{4}{x} = 0$

Step 2.5.1.4

Reorder terms.

$\frac{- 1 (x - 3)}{x (x - 3)} + \frac{4}{x} = 0$

Step 2.5.1.5

Cancel the common factor.

$\frac{- 1 (x - 3)}{x (x - 3)} + \frac{4}{x} = 0$

Step 2.5.1.6

Rewrite the expression.

$\frac{- 1}{x} + \frac{4}{x} = 0$

Step 2.5.2

Move the negative in front of the fraction.

$- \frac{1}{x} + \frac{4}{x} = 0$

Step 2.6

Combine the numerators over the common denominator.

$\frac{- 1 + 4}{x} = 0$

Step 2.7

Add $- 1$ and $4$ .

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

Step 3

Set the numerator equal to zero.

$3 = 0$

Step 4

Since $3 \neq 0$ , there are no solutions.

No solution