Basic Math Examples

$- \frac{5}{a - 2} + \frac{2 a + 1}{a}$

Step 1

To write $- \frac{5}{a - 2}$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

$- \frac{5}{a - 2} \cdot \frac{a}{a} + \frac{2 a + 1}{a}$

Step 2

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

$- \frac{5}{a - 2} \cdot \frac{a}{a} + \frac{2 a + 1}{a} \cdot \frac{a - 2}{a - 2}$

Step 3

Write each expression with a common denominator of $(a - 2) a$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{5}{a - 2}$ by $\frac{a}{a}$ .

$- \frac{5 a}{(a - 2) a} + \frac{2 a + 1}{a} \cdot \frac{a - 2}{a - 2}$

Step 3.2

Multiply $\frac{2 a + 1}{a}$ by $\frac{a - 2}{a - 2}$ .

$- \frac{5 a}{(a - 2) a} + \frac{(2 a + 1) (a - 2)}{a (a - 2)}$

Step 3.3

Reorder the factors of $(a - 2) a$ .

$- \frac{5 a}{a (a - 2)} + \frac{(2 a + 1) (a - 2)}{a (a - 2)}$

Step 4

Combine the numerators over the common denominator.

$\frac{- 5 a + (2 a + 1) (a - 2)}{a (a - 2)}$

Step 5

Simplify the numerator.

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Step 5.1

Expand $(2 a + 1) (a - 2)$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

$\frac{- 5 a + 2 a (a - 2) + 1 (a - 2)}{a (a - 2)}$

Step 5.1.2

Apply the distributive property.

$\frac{- 5 a + 2 a \cdot a + 2 a \cdot - 2 + 1 (a - 2)}{a (a - 2)}$

Step 5.1.3

Apply the distributive property.

$\frac{- 5 a + 2 a \cdot a + 2 a \cdot - 2 + 1 a + 1 \cdot - 2}{a (a - 2)}$

Step 5.2

Simplify and combine like terms.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $a$ by $a$ by adding the exponents.

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Step 5.2.1.1.1

Move $a$ .

$\frac{- 5 a + 2 (a \cdot a) + 2 a \cdot - 2 + 1 a + 1 \cdot - 2}{a (a - 2)}$

Step 5.2.1.1.2

Multiply $a$ by $a$ .

$\frac{- 5 a + 2 a^{2} + 2 a \cdot - 2 + 1 a + 1 \cdot - 2}{a (a - 2)}$

Step 5.2.1.2

Multiply $- 2$ by $2$ .

$\frac{- 5 a + 2 a^{2} - 4 a + 1 a + 1 \cdot - 2}{a (a - 2)}$

Step 5.2.1.3

Multiply $a$ by $1$ .

$\frac{- 5 a + 2 a^{2} - 4 a + a + 1 \cdot - 2}{a (a - 2)}$

Step 5.2.1.4

Multiply $- 2$ by $1$ .

$\frac{- 5 a + 2 a^{2} - 4 a + a - 2}{a (a - 2)}$

Step 5.2.2

Add $- 4 a$ and $a$ .

$\frac{- 5 a + 2 a^{2} - 3 a - 2}{a (a - 2)}$

Step 5.3

Subtract $3 a$ from $- 5 a$ .

$\frac{2 a^{2} - 8 a - 2}{a (a - 2)}$

Step 5.4

Factor $2$ out of $2 a^{2} - 8 a - 2$ .

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Step 5.4.1

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

$\frac{2 (a^{2}) - 8 a - 2}{a (a - 2)}$

Step 5.4.2

Factor $2$ out of $- 8 a$ .

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

Step 5.4.3

Factor $2$ out of $- 2$ .

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

Step 5.4.4

Factor $2$ out of $2 (a^{2}) + 2 (- 4 a)$ .

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

Step 5.4.5

Factor $2$ out of $2 (a^{2} - 4 a) + 2 (- 1)$ .

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