Simplify (n-4)/(7n^2-28n)+(3n)/(n-2)

$\frac{n - 4}{7 n^{2} - 28 n} + \frac{3 n}{n - 2}$

Simplify each term.

Factor $7 n$ out of $7 n^{2} - 28 n$ .

Factor $7 n$ out of $7 n^{2}$ .

$\frac{n - 4}{7 n (n) - 28 n} + \frac{3 n}{n - 2}$

Factor $7 n$ out of $- 28 n$ .

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

Factor $7 n$ out of $7 n (n) + 7 n (- 4)$ .

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

Cancel the common factor of $n - 4$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{7 n} + \frac{3 n}{n - 2}$

To write $\frac{1}{7 n}$ as a fraction with a common denominator, multiply by $\frac{n - 2}{n - 2}$ .

$\frac{1}{7 n} \cdot \frac{n - 2}{n - 2} + \frac{3 n}{n - 2}$

To write $\frac{3 n}{n - 2}$ as a fraction with a common denominator, multiply by $\frac{7 n}{7 n}$ .

$\frac{1}{7 n} \cdot \frac{n - 2}{n - 2} + \frac{3 n}{n - 2} \cdot \frac{7 n}{7 n}$

Write each expression with a common denominator of $7 n (n - 2)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{7 n}$ by $\frac{n - 2}{n - 2}$ .

$\frac{n - 2}{7 n (n - 2)} + \frac{3 n}{n - 2} \cdot \frac{7 n}{7 n}$

Multiply $\frac{3 n}{n - 2}$ by $\frac{7 n}{7 n}$ .

$\frac{n - 2}{7 n (n - 2)} + \frac{3 n (7 n)}{(n - 2) (7 n)}$

Reorder the factors of $(n - 2) (7 n)$ .

$\frac{n - 2}{7 n (n - 2)} + \frac{3 n (7 n)}{7 n (n - 2)}$

Combine the numerators over the common denominator.

$\frac{n - 2 + 3 n (7 n)}{7 n (n - 2)}$

Simplify the numerator.

Rewrite using the commutative property of multiplication.

$\frac{n - 2 + 3 \cdot 7 n \cdot n}{7 n (n - 2)}$

Multiply $n$ by $n$ by adding the exponents.

Move $n$ .

$\frac{n - 2 + 3 \cdot 7 (n \cdot n)}{7 n (n - 2)}$

Multiply $n$ by $n$ .

$\frac{n - 2 + 3 \cdot 7 n^{2}}{7 n (n - 2)}$

Multiply $3$ by $7$ .

$\frac{n - 2 + 21 n^{2}}{7 n (n - 2)}$

Reorder terms.

$\frac{21 n^{2} + n - 2}{7 n (n - 2)}$

Factor by grouping.

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 $a \cdot c = 21 \cdot - 2 = - 42$ and whose sum is $b = 1$ .

Multiply by $1$ .

$\frac{21 n^{2} + 1 n - 2}{7 n (n - 2)}$

Rewrite $1$ as $- 6$ plus $7$

$\frac{21 n^{2} + (- 6 + 7) n - 2}{7 n (n - 2)}$

Apply the distributive property.

$\frac{21 n^{2} - 6 n + 7 n - 2}{7 n (n - 2)}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(21 n^{2} - 6 n) + 7 n - 2}{7 n (n - 2)}$

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

$\frac{3 n (7 n - 2) + 1 (7 n - 2)}{7 n (n - 2)}$

Factor the polynomial by factoring out the greatest common factor, $7 n - 2$ .

$\frac{(7 n - 2) (3 n + 1)}{7 n (n - 2)}$

Simplify (n-4)/(7n^2-28n)+(3n)/(n-2)

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