Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Write each expression with a common denominator of $2 x^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

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

Step 3.2

Raise $x$ to the power of $1$ .

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

Step 3.3

Raise $x$ to the power of $1$ .

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

Step 3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.5

Add $1$ and $1$ .

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

Step 3.6

Multiply $\frac{x + 2}{x^{2}}$ by $\frac{2}{2}$ .

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

Step 3.7

Reorder the factors of $x^{2} \cdot 2$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $x$ by $1$ .

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

Step 5.3

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

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Step 5.3.1

Move $x$ .

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

Step 5.3.2

Multiply $x$ by $x$ .

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

Step 5.4

Apply the distributive property.

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

Step 5.5

Move $2$ to the left of $x$ .

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

Step 5.6

Multiply $2$ by $2$ .

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

Step 5.7

Add $x$ and $2 x$ .

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

Step 5.8

Factor by grouping.

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Step 5.8.1

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 $a \cdot c = - 1 \cdot 4 = - 4$ and whose sum is $b = 3$ .

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Step 5.8.1.1

Factor $3$ out of $3 x$ .

$\frac{- x^{2} + 3 (x) + 4}{2 x^{2}} + \frac{1}{3 a x^{2}}$

Step 5.8.1.2

Rewrite $3$ as $- 1$ plus $4$

$\frac{- x^{2} + (- 1 + 4) x + 4}{2 x^{2}} + \frac{1}{3 a x^{2}}$

Step 5.8.1.3

Apply the distributive property.

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

Step 5.8.2

Factor out the greatest common factor from each group.

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Step 5.8.2.1

Group the first two terms and the last two terms.

$\frac{(- x^{2} - 1 x) + 4 x + 4}{2 x^{2}} + \frac{1}{3 a x^{2}}$

Step 5.8.2.2

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

$\frac{x (- x - 1) - 4 (- x - 1)}{2 x^{2}} + \frac{1}{3 a x^{2}}$

Step 5.8.3

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

$\frac{(- x - 1) (x - 4)}{2 x^{2}} + \frac{1}{3 a x^{2}}$

Step 6

To write $\frac{(- x - 1) (x - 4)}{2 x^{2}}$ as a fraction with a common denominator, multiply by $\frac{3 a}{3 a}$ .

$\frac{(- x - 1) (x - 4)}{2 x^{2}} \cdot \frac{3 a}{3 a} + \frac{1}{3 a x^{2}}$

Step 7

To write $\frac{1}{3 a x^{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{(- x - 1) (x - 4)}{2 x^{2}} \cdot \frac{3 a}{3 a} + \frac{1}{3 a x^{2}} \cdot \frac{2}{2}$

Step 8

Write each expression with a common denominator of $6 a x^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.1

Multiply $\frac{(- x - 1) (x - 4)}{2 x^{2}}$ by $\frac{3 a}{3 a}$ .

$\frac{(- x - 1) (x - 4) (3 a)}{2 x^{2} (3 a)} + \frac{1}{3 a x^{2}} \cdot \frac{2}{2}$

Step 8.2

Multiply $3$ by $2$ .

$\frac{(- x - 1) (x - 4) (3 a)}{6 x^{2} a} + \frac{1}{3 a x^{2}} \cdot \frac{2}{2}$

Step 8.3

Multiply $\frac{1}{3 a x^{2}}$ by $\frac{2}{2}$ .

$\frac{(- x - 1) (x - 4) (3 a)}{6 x^{2} a} + \frac{2}{3 a x^{2} \cdot 2}$

Step 8.4

Multiply $2$ by $3$ .

$\frac{(- x - 1) (x - 4) (3 a)}{6 x^{2} a} + \frac{2}{6 a x^{2}}$

Step 8.5

Reorder the factors of $6 a x^{2}$ .

$\frac{(- x - 1) (x - 4) (3 a)}{6 x^{2} a} + \frac{2}{6 x^{2} a}$

Step 9

Combine the numerators over the common denominator.

$\frac{(- x - 1) (x - 4) (3 a) + 2}{6 x^{2} a}$

Step 10

Simplify the numerator.

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Step 10.1

Expand $(- x - 1) (x - 4)$ using the FOIL Method.

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Step 10.1.1

Apply the distributive property.

$\frac{(- x (x - 4) - 1 (x - 4)) \cdot 3 a + 2}{6 x^{2} a}$

Step 10.1.2

Apply the distributive property.

$\frac{(- x \cdot x - x \cdot - 4 - 1 (x - 4)) \cdot 3 a + 2}{6 x^{2} a}$

Step 10.1.3

Apply the distributive property.

$\frac{(- x \cdot x - x \cdot - 4 - 1 x - 1 \cdot - 4) \cdot 3 a + 2}{6 x^{2} a}$

Step 10.2

Simplify and combine like terms.

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Step 10.2.1

Simplify each term.

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Step 10.2.1.1

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

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Step 10.2.1.1.1

Move $x$ .

$\frac{(- (x \cdot x) - x \cdot - 4 - 1 x - 1 \cdot - 4) \cdot 3 a + 2}{6 x^{2} a}$

Step 10.2.1.1.2

Multiply $x$ by $x$ .

$\frac{(- x^{2} - x \cdot - 4 - 1 x - 1 \cdot - 4) \cdot 3 a + 2}{6 x^{2} a}$

Step 10.2.1.2

Multiply $- 4$ by $- 1$ .

$\frac{(- x^{2} + 4 x - 1 x - 1 \cdot - 4) \cdot 3 a + 2}{6 x^{2} a}$

Step 10.2.1.3

Rewrite $- 1 x$ as $- x$ .

$\frac{(- x^{2} + 4 x - x - 1 \cdot - 4) \cdot 3 a + 2}{6 x^{2} a}$

Step 10.2.1.4

Multiply $- 1$ by $- 4$ .

$\frac{(- x^{2} + 4 x - x + 4) \cdot 3 a + 2}{6 x^{2} a}$

Step 10.2.2

Subtract $x$ from $4 x$ .

$\frac{(- x^{2} + 3 x + 4) \cdot 3 a + 2}{6 x^{2} a}$

Step 10.3

Apply the distributive property.

$\frac{(- x^{2} \cdot 3 + 3 x \cdot 3 + 4 \cdot 3) a + 2}{6 x^{2} a}$

Step 10.4

Simplify.

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Step 10.4.1

Multiply $3$ by $- 1$ .

$\frac{(- 3 x^{2} + 3 x \cdot 3 + 4 \cdot 3) a + 2}{6 x^{2} a}$

Step 10.4.2

Multiply $3$ by $3$ .

$\frac{(- 3 x^{2} + 9 x + 4 \cdot 3) a + 2}{6 x^{2} a}$

Step 10.4.3

Multiply $4$ by $3$ .

$\frac{(- 3 x^{2} + 9 x + 12) a + 2}{6 x^{2} a}$

Step 10.5

Apply the distributive property.

$\frac{- 3 x^{2} a + 9 x a + 12 a + 2}{6 x^{2} a}$

Step 11

Simplify with factoring out.

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Step 11.1

Factor $- 1$ out of $- 3 x^{2} a$ .

$\frac{- (3 x^{2} a) + 9 x a + 12 a + 2}{6 x^{2} a}$

Step 11.2

Factor $- 1$ out of $9 x a$ .

$\frac{- (3 x^{2} a) - (- 9 x a) + 12 a + 2}{6 x^{2} a}$

Step 11.3

Factor $- 1$ out of $- (3 x^{2} a) - (- 9 x a)$ .

$\frac{- (3 x^{2} a - 9 x a) + 12 a + 2}{6 x^{2} a}$

Step 11.4

Factor $- 1$ out of $12 a$ .

$\frac{- (3 x^{2} a - 9 x a) - (- 12 a) + 2}{6 x^{2} a}$

Step 11.5

Factor $- 1$ out of $- (3 x^{2} a - 9 x a) - (- 12 a)$ .

$\frac{- (3 x^{2} a - 9 x a - 12 a) + 2}{6 x^{2} a}$

Step 11.6

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{- (3 x^{2} a - 9 x a - 12 a) - 1 (- 2)}{6 x^{2} a}$

Step 11.7

Factor $- 1$ out of $- (3 x^{2} a - 9 x a - 12 a) - 1 (- 2)$ .

$\frac{- (3 x^{2} a - 9 x a - 12 a - 2)}{6 x^{2} a}$

Step 11.8

Simplify the expression.

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Step 11.8.1

Rewrite $- (3 x^{2} a - 9 x a - 12 a - 2)$ as $- 1 (3 x^{2} a - 9 x a - 12 a - 2)$ .

$\frac{- 1 (3 x^{2} a - 9 x a - 12 a - 2)}{6 x^{2} a}$

Step 11.8.2

Move the negative in front of the fraction.

$- \frac{3 x^{2} a - 9 x a - 12 a - 2}{6 x^{2} a}$