Algebra Examples

$\frac{X}{X - 1} - \frac{3}{X} = \frac{1}{X^{2} - X}$

Step 1

Subtract $\frac{1}{X^{2} - X}$ from both sides of the equation.

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

Step 2

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

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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 $X^{2}$ .

$\frac{X}{X - 1} - \frac{3}{X} - \frac{1}{X \cdot X - X} = 0$

Step 2.1.2

Factor $X$ out of $- X$ .

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

Step 2.1.3

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

$\frac{X}{X - 1} - \frac{3}{X} - \frac{1}{X (X - 1)} = 0$

Step 2.2

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

$\frac{X}{X - 1} \cdot \frac{X}{X} - \frac{3}{X} - \frac{1}{X (X - 1)} = 0$

Step 2.3

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

$\frac{X}{X - 1} \cdot \frac{X}{X} - \frac{3}{X} \cdot \frac{X - 1}{X - 1} - \frac{1}{X (X - 1)} = 0$

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 $\frac{X}{X - 1}$ by $\frac{X}{X}$ .

$\frac{X \cdot X}{(X - 1) X} - \frac{3}{X} \cdot \frac{X - 1}{X - 1} - \frac{1}{X (X - 1)} = 0$

Step 2.4.2

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

$\frac{X \cdot X}{(X - 1) X} - \frac{3 (X - 1)}{X (X - 1)} - \frac{1}{X (X - 1)} = 0$

Step 2.4.3

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

$\frac{X \cdot X}{X (X - 1)} - \frac{3 (X - 1)}{X (X - 1)} - \frac{1}{X (X - 1)} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{X \cdot X - 3 (X - 1)}{X (X - 1)} - \frac{1}{X (X - 1)} = 0$

Step 2.6

Combine the numerators over the common denominator.

$\frac{X \cdot X - 3 (X - 1) - 1}{X (X - 1)} = 0$

Step 2.7

Simplify each term.

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Step 2.7.1

Multiply $X$ by $X$ .

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

Step 2.7.2

Apply the distributive property.

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

Step 2.7.3

Multiply $- 3$ by $- 1$ .

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

Step 2.8

Subtract $1$ from $3$ .

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

Step 2.9

Factor $X^{2} - 3 X + 2$ using the AC method.

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Step 2.9.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $- 3$ .

$- 2, - 1$

Step 2.9.2

Write the factored form using these integers.

$\frac{(X - 2) (X - 1)}{X (X - 1)} = 0$

Step 2.10

Cancel the common factor of $X - 1$ .

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Step 2.10.1

Cancel the common factor.

$\frac{(X - 2) (X - 1)}{X (X - 1)} = 0$

Step 2.10.2

Rewrite the expression.

$\frac{X - 2}{X} = 0$

Step 3

Set the numerator equal to zero.

$X - 2 = 0$

Step 4

Add $2$ to both sides of the equation.

$X = 2$