Basic Math Examples

$\frac{2}{3} + \frac{n - 2}{10 n^{2} + 8 n}$

Step 1

To write $\frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{10 n^{2} + 8 n}{10 n^{2} + 8 n}$ .

$\frac{2}{3} \cdot \frac{10 n^{2} + 8 n}{10 n^{2} + 8 n} + \frac{n - 2}{10 n^{2} + 8 n}$

Step 2

To write $\frac{n - 2}{10 n^{2} + 8 n}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2}{3} \cdot \frac{10 n^{2} + 8 n}{10 n^{2} + 8 n} + \frac{n - 2}{10 n^{2} + 8 n} \cdot \frac{3}{3}$

Step 3

Write each expression with a common denominator of $3 (10 n^{2} + 8 n)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{2}{3}$ by $\frac{10 n^{2} + 8 n}{10 n^{2} + 8 n}$ .

$\frac{2 (10 n^{2} + 8 n)}{3 (10 n^{2} + 8 n)} + \frac{n - 2}{10 n^{2} + 8 n} \cdot \frac{3}{3}$

Step 3.2

Multiply $\frac{n - 2}{10 n^{2} + 8 n}$ by $\frac{3}{3}$ .

$\frac{2 (10 n^{2} + 8 n)}{3 (10 n^{2} + 8 n)} + \frac{(n - 2) \cdot 3}{(10 n^{2} + 8 n) \cdot 3}$

Step 3.3

Reorder the factors of $(10 n^{2} + 8 n) \cdot 3$ .

$\frac{2 (10 n^{2} + 8 n)}{3 (10 n^{2} + 8 n)} + \frac{(n - 2) \cdot 3}{3 (10 n^{2} + 8 n)}$

Step 4

Combine the numerators over the common denominator.

$\frac{2 (10 n^{2} + 8 n) + (n - 2) \cdot 3}{3 (10 n^{2} + 8 n)}$

Step 5

Reorder terms.

$\frac{3 (n - 2) + 2 (10 n^{2} + 8 n)}{3 (10 n^{2} + 8 n)}$

Step 6

Rewrite $\frac{3 (n - 2) + 2 (10 n^{2} + 8 n)}{3 (10 n^{2} + 8 n)}$ in a factored form.

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Step 6.1

Apply the distributive property.

$\frac{3 n + 3 \cdot - 2 + 2 (10 n^{2} + 8 n)}{3 (10 n^{2} + 8 n)}$

Step 6.2

Multiply $3$ by $- 2$ .

$\frac{3 n - 6 + 2 (10 n^{2} + 8 n)}{3 (10 n^{2} + 8 n)}$

Step 6.3

Apply the distributive property.

$\frac{3 n - 6 + 2 (10 n^{2}) + 2 (8 n)}{3 (10 n^{2} + 8 n)}$

Step 6.4

Multiply $10$ by $2$ .

$\frac{3 n - 6 + 20 n^{2} + 2 (8 n)}{3 (10 n^{2} + 8 n)}$

Step 6.5

Multiply $8$ by $2$ .

$\frac{3 n - 6 + 20 n^{2} + 16 n}{3 (10 n^{2} + 8 n)}$

Step 6.6

Add $3 n$ and $16 n$ .

$\frac{19 n - 6 + 20 n^{2}}{3 (10 n^{2} + 8 n)}$

Step 6.7

Reorder terms.

$\frac{20 n^{2} + 19 n - 6}{3 (10 n^{2} + 8 n)}$

Step 6.8

Factor by grouping.

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Step 6.8.1

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 $a \cdot c = 20 \cdot - 6 = - 120$ and whose sum is $b = 19$ .

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Step 6.8.1.1

Factor $19$ out of $19 n$ .

$\frac{20 n^{2} + 19 (n) - 6}{3 (10 n^{2} + 8 n)}$

Step 6.8.1.2

Rewrite $19$ as $- 5$ plus $24$

$\frac{20 n^{2} + (- 5 + 24) n - 6}{3 (10 n^{2} + 8 n)}$

Step 6.8.1.3

Apply the distributive property.

$\frac{20 n^{2} - 5 n + 24 n - 6}{3 (10 n^{2} + 8 n)}$

Step 6.8.2

Factor out the greatest common factor from each group.

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Step 6.8.2.1

Group the first two terms and the last two terms.

$\frac{(20 n^{2} - 5 n) + 24 n - 6}{3 (10 n^{2} + 8 n)}$

Step 6.8.2.2

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

$\frac{5 n (4 n - 1) + 6 (4 n - 1)}{3 (10 n^{2} + 8 n)}$

Step 6.8.3

Factor the polynomial by factoring out the greatest common factor, $4 n - 1$ .

$\frac{(4 n - 1) (5 n + 6)}{3 (10 n^{2} + 8 n)}$

Step 6.9

Factor.

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Step 6.9.1

Factor $2 n$ out of $10 n^{2} + 8 n$ .

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Step 6.9.1.1

Factor $2 n$ out of $10 n^{2}$ .

$\frac{(4 n - 1) (5 n + 6)}{3 (2 n (5 n) + 8 n)}$

Step 6.9.1.2

Factor $2 n$ out of $8 n$ .

$\frac{(4 n - 1) (5 n + 6)}{3 (2 n (5 n) + 2 n (4))}$

Step 6.9.1.3

Factor $2 n$ out of $2 n (5 n) + 2 n (4)$ .

$\frac{(4 n - 1) (5 n + 6)}{3 (2 n (5 n + 4))}$

Step 6.9.2

Remove unnecessary parentheses.

$\frac{(4 n - 1) (5 n + 6)}{3 \cdot 2 n (5 n + 4)}$

Step 6.10

Multiply $3$ by $2$ .

$\frac{(4 n - 1) (5 n + 6)}{6 n (5 n + 4)}$