Algebra Examples

$\frac{2 k}{3 k + 12} + \frac{k + 7}{k^{2} + 4 k}$

Step 1

Simplify each term.

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Step 1.1

Factor $3$ out of $3 k + 12$ .

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Step 1.1.1

Factor $3$ out of $3 k$ .

$\frac{2 k}{3 (k) + 12} + \frac{k + 7}{k^{2} + 4 k}$

Step 1.1.2

Factor $3$ out of $12$ .

$\frac{2 k}{3 k + 3 \cdot 4} + \frac{k + 7}{k^{2} + 4 k}$

Step 1.1.3

Factor $3$ out of $3 k + 3 \cdot 4$ .

$\frac{2 k}{3 (k + 4)} + \frac{k + 7}{k^{2} + 4 k}$

Step 1.2

Factor $k$ out of $k^{2} + 4 k$ .

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Step 1.2.1

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

$\frac{2 k}{3 (k + 4)} + \frac{k + 7}{k \cdot k + 4 k}$

Step 1.2.2

Factor $k$ out of $4 k$ .

$\frac{2 k}{3 (k + 4)} + \frac{k + 7}{k \cdot k + k \cdot 4}$

Step 1.2.3

Factor $k$ out of $k \cdot k + k \cdot 4$ .

$\frac{2 k}{3 (k + 4)} + \frac{k + 7}{k (k + 4)}$

Step 2

To write $\frac{2 k}{3 (k + 4)}$ as a fraction with a common denominator, multiply by $\frac{k}{k}$ .

$\frac{2 k}{3 (k + 4)} \cdot \frac{k}{k} + \frac{k + 7}{k (k + 4)}$

Step 3

To write $\frac{k + 7}{k (k + 4)}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 k}{3 (k + 4)} \cdot \frac{k}{k} + \frac{k + 7}{k (k + 4)} \cdot \frac{3}{3}$

Step 4

Write each expression with a common denominator of $3 (k + 4) k$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{2 k}{3 (k + 4)}$ by $\frac{k}{k}$ .

$\frac{2 k \cdot k}{3 (k + 4) k} + \frac{k + 7}{k (k + 4)} \cdot \frac{3}{3}$

Step 4.2

Multiply $\frac{k + 7}{k (k + 4)}$ by $\frac{3}{3}$ .

$\frac{2 k \cdot k}{3 (k + 4) k} + \frac{(k + 7) \cdot 3}{k (k + 4) \cdot 3}$

Step 4.3

Reorder the factors of $3 (k + 4) k$ .

$\frac{2 k \cdot k}{3 k (k + 4)} + \frac{(k + 7) \cdot 3}{k (k + 4) \cdot 3}$

Step 4.4

Reorder the factors of $k (k + 4) \cdot 3$ .

$\frac{2 k \cdot k}{3 k (k + 4)} + \frac{(k + 7) \cdot 3}{3 k (k + 4)}$

Step 5

Combine the numerators over the common denominator.

$\frac{2 k \cdot k + (k + 7) \cdot 3}{3 k (k + 4)}$

Step 6

Simplify the numerator.

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Step 6.1

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

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Step 6.1.1

Move $k$ .

$\frac{2 (k \cdot k) + (k + 7) \cdot 3}{3 k (k + 4)}$

Step 6.1.2

Multiply $k$ by $k$ .

$\frac{2 k^{2} + (k + 7) \cdot 3}{3 k (k + 4)}$

Step 6.2

Apply the distributive property.

$\frac{2 k^{2} + k \cdot 3 + 7 \cdot 3}{3 k (k + 4)}$

Step 6.3

Move $3$ to the left of $k$ .

$\frac{2 k^{2} + 3 \cdot k + 7 \cdot 3}{3 k (k + 4)}$

Step 6.4

Multiply $7$ by $3$ .

$\frac{2 k^{2} + 3 k + 21}{3 k (k + 4)}$