Algebra Examples

$\frac{a - \frac{2 a - 1}{a}}{\frac{1 - a}{3 a}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$(a - \frac{2 a - 1}{a}) \frac{3 a}{1 - a}$

Step 2

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

$(\frac{a \cdot a}{a} - \frac{2 a - 1}{a}) \frac{3 a}{1 - a}$

Step 3

Combine the numerators over the common denominator.

$\frac{a \cdot a - (2 a - 1)}{a} \cdot \frac{3 a}{1 - a}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $a$ by $a$ .

$\frac{a^{2} - (2 a - 1)}{a} \cdot \frac{3 a}{1 - a}$

Step 4.2

Apply the distributive property.

$\frac{a^{2} - (2 a) - - 1}{a} \cdot \frac{3 a}{1 - a}$

Step 4.3

Multiply $2$ by $- 1$ .

$\frac{a^{2} - 2 a - - 1}{a} \cdot \frac{3 a}{1 - a}$

Step 4.4

Multiply $- 1$ by $- 1$ .

$\frac{a^{2} - 2 a + 1}{a} \cdot \frac{3 a}{1 - a}$

Step 4.5

Factor using the perfect square rule.

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Step 4.5.1

Rewrite $1$ as $1^{2}$ .

$\frac{a^{2} - 2 a + 1^{2}}{a} \cdot \frac{3 a}{1 - a}$

Step 4.5.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 a = 2 \cdot a \cdot 1$

Step 4.5.3

Rewrite the polynomial.

$\frac{a^{2} - 2 \cdot a \cdot 1 + 1^{2}}{a} \cdot \frac{3 a}{1 - a}$

Step 4.5.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = a$ and $b = 1$ .

$\frac{{(a - 1)}^{2}}{a} \cdot \frac{3 a}{1 - a}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $a$ .

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Step 5.1.1

Factor $a$ out of $3 a$ .

$\frac{{(a - 1)}^{2}}{a} \cdot \frac{a \cdot 3}{1 - a}$

Step 5.1.2

Cancel the common factor.

$\frac{{(a - 1)}^{2}}{a} \cdot \frac{a \cdot 3}{1 - a}$

Step 5.1.3

Rewrite the expression.

${(a - 1)}^{2} \frac{3}{1 - a}$

Step 5.2

Combine ${(a - 1)}^{2}$ and $\frac{3}{1 - a}$ .

$\frac{{(a - 1)}^{2} \cdot 3}{1 - a}$

Step 5.3

Cancel the common factor of ${(a - 1)}^{2}$ and $1 - a$ .

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Step 5.3.1

Factor $- 1$ out of $a$ .

$\frac{{(- 1 (- a) - 1)}^{2} \cdot 3}{1 - a}$

Step 5.3.2

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

$\frac{{(- 1 (- a) - 1 (1))}^{2} \cdot 3}{1 - a}$

Step 5.3.3

Factor $- 1$ out of $- 1 (- a) - 1 (1)$ .

$\frac{{(- 1 (- a + 1))}^{2} \cdot 3}{1 - a}$

Step 5.3.4

Apply the product rule to $- 1 (- a + 1)$ .

$\frac{{(- 1)}^{2} {(- a + 1)}^{2} \cdot 3}{1 - a}$

Step 5.3.5

Raise $- 1$ to the power of $2$ .

$\frac{1 {(- a + 1)}^{2} \cdot 3}{1 - a}$

Step 5.3.6

Multiply ${(- a + 1)}^{2}$ by $1$ .

$\frac{{(- a + 1)}^{2} \cdot 3}{1 - a}$

Step 5.3.7

Reorder terms.

$\frac{{(- a + 1)}^{2} \cdot 3}{- a + 1}$

Step 5.3.8

Factor $- a + 1$ out of ${(- a + 1)}^{2} \cdot 3$ .

$\frac{(- a + 1) ((- a + 1) \cdot 3)}{- a + 1}$

Step 5.3.9

Cancel the common factors.

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Step 5.3.9.1

Multiply by $1$ .

$\frac{(- a + 1) ((- a + 1) \cdot 3)}{(- a + 1) \cdot 1}$

Step 5.3.9.2

Cancel the common factor.

$\frac{(- a + 1) ((- a + 1) \cdot 3)}{(- a + 1) \cdot 1}$

Step 5.3.9.3

Rewrite the expression.

$\frac{(- a + 1) \cdot 3}{1}$

Step 5.3.9.4

Divide $(- a + 1) \cdot 3$ by $1$ .

$(- a + 1) \cdot 3$

Step 5.4

Apply the distributive property.

$- a \cdot 3 + 1 \cdot 3$

Step 5.5

Multiply.

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Step 5.5.1

Multiply $3$ by $- 1$ .

$- 3 a + 1 \cdot 3$

Step 5.5.2

Multiply $3$ by $1$ .

$- 3 a + 3$