Algebra Examples

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

Step 1

Subtract $\frac{3}{x^{2} - x}$ from both sides of the equation.

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

Step 2

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

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Factor $x$ out of $- x$ .

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

Step 2.1.3

Factor $x$ out of $x \cdot x + x \cdot - 1$ .

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

Step 2.2

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

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

Step 2.3

To write $\frac{2}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

Multiply $\frac{2}{x}$ by $\frac{x - 1}{x - 1}$ .

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

Step 2.4.3

Reorder the factors of $(x - 1) x$ .

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 2.7

Simplify each term.

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Step 2.7.1

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

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Step 2.7.1.1

Move $x$ .

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

Step 2.7.1.2

Multiply $x$ by $x$ .

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

Step 2.7.2

Apply the distributive property.

$\frac{3 x^{2} + 2 x + 2 \cdot - 1 - 3}{x (x - 1)} = 0$

Step 2.7.3

Multiply $2$ by $- 1$ .

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

Step 2.8

Subtract $3$ from $- 2$ .

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

Step 2.9

Factor by grouping.

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Step 2.9.1

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 $a \cdot c = 3 \cdot - 5 = - 15$ and whose sum is $b = 2$ .

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Step 2.9.1.1

Factor $2$ out of $2 x$ .

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

Step 2.9.1.2

Rewrite $2$ as $- 3$ plus $5$

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

Step 2.9.1.3

Apply the distributive property.

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

Step 2.9.2

Factor out the greatest common factor from each group.

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Step 2.9.2.1

Group the first two terms and the last two terms.

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

Step 2.9.2.2

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

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

Step 2.9.3

Factor the polynomial by factoring out the greatest common factor, $x - 1$ .

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

Step 2.10

Cancel the common factor of $x - 1$ .

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Step 2.10.1

Cancel the common factor.

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

Step 2.10.2

Rewrite the expression.

$\frac{3 x + 5}{x} = 0$

Step 3

Set the numerator equal to zero.

$3 x + 5 = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Subtract $5$ from both sides of the equation.

$3 x = - 5$

Step 4.2

Divide each term in $3 x = - 5$ by $3$ and simplify.

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Step 4.2.1

Divide each term in $3 x = - 5$ by $3$ .

$\frac{3 x}{3} = \frac{- 5}{3}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 5}{3}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{3}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Move the negative in front of the fraction.

$x = - \frac{5}{3}$