Basic Math Examples

$\frac{k}{k - 5} + \frac{k - 1}{k + 5}$

Step 1

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

$\frac{k}{k - 5} \cdot \frac{k + 5}{k + 5} + \frac{k - 1}{k + 5}$

Step 2

To write $\frac{k - 1}{k + 5}$ as a fraction with a common denominator, multiply by $\frac{k - 5}{k - 5}$ .

$\frac{k}{k - 5} \cdot \frac{k + 5}{k + 5} + \frac{k - 1}{k + 5} \cdot \frac{k - 5}{k - 5}$

Step 3

Write each expression with a common denominator of $(k - 5) (k + 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

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

Step 3.2

Multiply $\frac{k - 1}{k + 5}$ by $\frac{k - 5}{k - 5}$ .

$\frac{k (k + 5)}{(k - 5) (k + 5)} + \frac{(k - 1) (k - 5)}{(k + 5) (k - 5)}$

Step 3.3

Reorder the factors of $(k - 5) (k + 5)$ .

$\frac{k (k + 5)}{(k + 5) (k - 5)} + \frac{(k - 1) (k - 5)}{(k + 5) (k - 5)}$

Step 4

Combine the numerators over the common denominator.

$\frac{k (k + 5) + (k - 1) (k - 5)}{(k + 5) (k - 5)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $k$ by $k$ .

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

Step 5.3

Move $5$ to the left of $k$ .

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

Step 5.4

Expand $(k - 1) (k - 5)$ using the FOIL Method.

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Step 5.4.1

Apply the distributive property.

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

Step 5.4.2

Apply the distributive property.

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

Step 5.4.3

Apply the distributive property.

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

Step 5.5

Simplify and combine like terms.

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Step 5.5.1

Simplify each term.

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Step 5.5.1.1

Multiply $k$ by $k$ .

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

Step 5.5.1.2

Move $- 5$ to the left of $k$ .

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

Step 5.5.1.3

Rewrite $- 1 k$ as $- k$ .

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

Step 5.5.1.4

Multiply $- 1$ by $- 5$ .

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

Step 5.5.2

Subtract $k$ from $- 5 k$ .

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

Step 5.6

Add $k^{2}$ and $k^{2}$ .

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

Step 5.7

Subtract $6 k$ from $5 k$ .

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