Basic Math Examples

$\frac{6}{k - 8} - \frac{k + 5}{k - 2}$

Step 1

To write $\frac{6}{k - 8}$ as a fraction with a common denominator, multiply by $\frac{k - 2}{k - 2}$ .

$\frac{6}{k - 8} \cdot \frac{k - 2}{k - 2} - \frac{k + 5}{k - 2}$

Step 2

To write $- \frac{k + 5}{k - 2}$ as a fraction with a common denominator, multiply by $\frac{k - 8}{k - 8}$ .

$\frac{6}{k - 8} \cdot \frac{k - 2}{k - 2} - \frac{k + 5}{k - 2} \cdot \frac{k - 8}{k - 8}$

Step 3

Write each expression with a common denominator of $(k - 8) (k - 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{6}{k - 8}$ by $\frac{k - 2}{k - 2}$ .

$\frac{6 (k - 2)}{(k - 8) (k - 2)} - \frac{k + 5}{k - 2} \cdot \frac{k - 8}{k - 8}$

Step 3.2

Multiply $\frac{k + 5}{k - 2}$ by $\frac{k - 8}{k - 8}$ .

$\frac{6 (k - 2)}{(k - 8) (k - 2)} - \frac{(k + 5) (k - 8)}{(k - 2) (k - 8)}$

Step 3.3

Reorder the factors of $(k - 8) (k - 2)$ .

$\frac{6 (k - 2)}{(k - 2) (k - 8)} - \frac{(k + 5) (k - 8)}{(k - 2) (k - 8)}$

Step 4

Combine the numerators over the common denominator.

$\frac{6 (k - 2) - (k + 5) (k - 8)}{(k - 2) (k - 8)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{6 k + 6 \cdot - 2 - (k + 5) (k - 8)}{(k - 2) (k - 8)}$

Step 5.2

Multiply $6$ by $- 2$ .

$\frac{6 k - 12 - (k + 5) (k - 8)}{(k - 2) (k - 8)}$

Step 5.3

Apply the distributive property.

$\frac{6 k - 12 + (- k - 1 \cdot 5) (k - 8)}{(k - 2) (k - 8)}$

Step 5.4

Multiply $- 1$ by $5$ .

$\frac{6 k - 12 + (- k - 5) (k - 8)}{(k - 2) (k - 8)}$

Step 5.5

Expand $(- k - 5) (k - 8)$ using the FOIL Method.

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Step 5.5.1

Apply the distributive property.

$\frac{6 k - 12 - k (k - 8) - 5 (k - 8)}{(k - 2) (k - 8)}$

Step 5.5.2

Apply the distributive property.

$\frac{6 k - 12 - k \cdot k - k \cdot - 8 - 5 (k - 8)}{(k - 2) (k - 8)}$

Step 5.5.3

Apply the distributive property.

$\frac{6 k - 12 - k \cdot k - k \cdot - 8 - 5 k - 5 \cdot - 8}{(k - 2) (k - 8)}$

Step 5.6

Simplify and combine like terms.

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Step 5.6.1

Simplify each term.

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Step 5.6.1.1

Multiply $k$ by $k$ by adding the exponents.

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Step 5.6.1.1.1

Move $k$ .

$\frac{6 k - 12 - (k \cdot k) - k \cdot - 8 - 5 k - 5 \cdot - 8}{(k - 2) (k - 8)}$

Step 5.6.1.1.2

Multiply $k$ by $k$ .

$\frac{6 k - 12 - k^{2} - k \cdot - 8 - 5 k - 5 \cdot - 8}{(k - 2) (k - 8)}$

Step 5.6.1.2

Multiply $- 8$ by $- 1$ .

$\frac{6 k - 12 - k^{2} + 8 k - 5 k - 5 \cdot - 8}{(k - 2) (k - 8)}$

Step 5.6.1.3

Multiply $- 5$ by $- 8$ .

$\frac{6 k - 12 - k^{2} + 8 k - 5 k + 40}{(k - 2) (k - 8)}$

Step 5.6.2

Subtract $5 k$ from $8 k$ .

$\frac{6 k - 12 - k^{2} + 3 k + 40}{(k - 2) (k - 8)}$

Step 5.7

Add $6 k$ and $3 k$ .

$\frac{9 k - 12 - k^{2} + 40}{(k - 2) (k - 8)}$

Step 5.8

Add $- 12$ and $40$ .

$\frac{9 k - k^{2} + 28}{(k - 2) (k - 8)}$

Step 5.9

Reorder terms.

$\frac{- k^{2} + 9 k + 28}{(k - 2) (k - 8)}$

Step 6

Simplify with factoring out.

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Step 6.1

Factor $- 1$ out of $- k^{2}$ .

$\frac{- (k^{2}) + 9 k + 28}{(k - 2) (k - 8)}$

Step 6.2

Factor $- 1$ out of $9 k$ .

$\frac{- (k^{2}) - (- 9 k) + 28}{(k - 2) (k - 8)}$

Step 6.3

Factor $- 1$ out of $- (k^{2}) - (- 9 k)$ .

$\frac{- (k^{2} - 9 k) + 28}{(k - 2) (k - 8)}$

Step 6.4

Rewrite $28$ as $- 1 (- 28)$ .

$\frac{- (k^{2} - 9 k) - 1 (- 28)}{(k - 2) (k - 8)}$

Step 6.5

Factor $- 1$ out of $- (k^{2} - 9 k) - 1 (- 28)$ .

$\frac{- (k^{2} - 9 k - 28)}{(k - 2) (k - 8)}$

Step 6.6

Simplify the expression.

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Step 6.6.1

Rewrite $- (k^{2} - 9 k - 28)$ as $- 1 (k^{2} - 9 k - 28)$ .

$\frac{- 1 (k^{2} - 9 k - 28)}{(k - 2) (k - 8)}$

Step 6.6.2

Move the negative in front of the fraction.

$- \frac{k^{2} - 9 k - 28}{(k - 2) (k - 8)}$