Basic Math Examples

$\frac{n^{2} + n - 20}{n^{2} + 2 n} - 15$

Step 1

Simplify each term.

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Step 1.1

Factor $n^{2} + n - 20$ using the AC method.

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Step 1.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 20$ and whose sum is $1$ .

$- 4, 5$

Step 1.1.2

Write the factored form using these integers.

$\frac{(n - 4) (n + 5)}{n^{2} + 2 n} - 15$

Step 1.2

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

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Step 1.2.1

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

$\frac{(n - 4) (n + 5)}{n \cdot n + 2 n} - 15$

Step 1.2.2

Factor $n$ out of $2 n$ .

$\frac{(n - 4) (n + 5)}{n \cdot n + n \cdot 2} - 15$

Step 1.2.3

Factor $n$ out of $n \cdot n + n \cdot 2$ .

$\frac{(n - 4) (n + 5)}{n (n + 2)} - 15$

Step 2

To write $- 15$ as a fraction with a common denominator, multiply by $\frac{n (n + 2)}{n (n + 2)}$ .

$\frac{(n - 4) (n + 5)}{n (n + 2)} - 15 \cdot \frac{n (n + 2)}{n (n + 2)}$

Step 3

Simplify terms.

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Step 3.1

Combine $- 15$ and $\frac{n (n + 2)}{n (n + 2)}$ .

$\frac{(n - 4) (n + 5)}{n (n + 2)} + \frac{- 15 (n (n + 2))}{n (n + 2)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{(n - 4) (n + 5) - 15 (n (n + 2))}{n (n + 2)}$

Step 4

Simplify the numerator.

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Step 4.1

Expand $(n - 4) (n + 5)$ using the FOIL Method.

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Step 4.1.1

Apply the distributive property.

$\frac{n (n + 5) - 4 (n + 5) - 15 n (n + 2)}{n (n + 2)}$

Step 4.1.2

Apply the distributive property.

$\frac{n \cdot n + n \cdot 5 - 4 (n + 5) - 15 n (n + 2)}{n (n + 2)}$

Step 4.1.3

Apply the distributive property.

$\frac{n \cdot n + n \cdot 5 - 4 n - 4 \cdot 5 - 15 n (n + 2)}{n (n + 2)}$

Step 4.2

Simplify and combine like terms.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $n$ by $n$ .

$\frac{n^{2} + n \cdot 5 - 4 n - 4 \cdot 5 - 15 n (n + 2)}{n (n + 2)}$

Step 4.2.1.2

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

$\frac{n^{2} + 5 \cdot n - 4 n - 4 \cdot 5 - 15 n (n + 2)}{n (n + 2)}$

Step 4.2.1.3

Multiply $- 4$ by $5$ .

$\frac{n^{2} + 5 n - 4 n - 20 - 15 n (n + 2)}{n (n + 2)}$

Step 4.2.2

Subtract $4 n$ from $5 n$ .

$\frac{n^{2} + n - 20 - 15 n (n + 2)}{n (n + 2)}$

Step 4.3

Apply the distributive property.

$\frac{n^{2} + n - 20 - 15 n \cdot n - 15 n \cdot 2}{n (n + 2)}$

Step 4.4

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

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Step 4.4.1

Move $n$ .

$\frac{n^{2} + n - 20 - 15 (n \cdot n) - 15 n \cdot 2}{n (n + 2)}$

Step 4.4.2

Multiply $n$ by $n$ .

$\frac{n^{2} + n - 20 - 15 n^{2} - 15 n \cdot 2}{n (n + 2)}$

Step 4.5

Multiply $2$ by $- 15$ .

$\frac{n^{2} + n - 20 - 15 n^{2} - 30 n}{n (n + 2)}$

Step 4.6

Subtract $15 n^{2}$ from $n^{2}$ .

$\frac{- 14 n^{2} + n - 20 - 30 n}{n (n + 2)}$

Step 4.7

Subtract $30 n$ from $n$ .

$\frac{- 14 n^{2} - 29 n - 20}{n (n + 2)}$

Step 5

Simplify with factoring out.

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Step 5.1

Factor $- 1$ out of $- 14 n^{2}$ .

$\frac{- (14 n^{2}) - 29 n - 20}{n (n + 2)}$

Step 5.2

Factor $- 1$ out of $- 29 n$ .

$\frac{- (14 n^{2}) - (29 n) - 20}{n (n + 2)}$

Step 5.3

Factor $- 1$ out of $- (14 n^{2}) - (29 n)$ .

$\frac{- (14 n^{2} + 29 n) - 20}{n (n + 2)}$

Step 5.4

Rewrite $- 20$ as $- 1 (20)$ .

$\frac{- (14 n^{2} + 29 n) - 1 (20)}{n (n + 2)}$

Step 5.5

Factor $- 1$ out of $- (14 n^{2} + 29 n) - 1 (20)$ .

$\frac{- (14 n^{2} + 29 n + 20)}{n (n + 2)}$

Step 5.6

Simplify the expression.

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Step 5.6.1

Rewrite $- (14 n^{2} + 29 n + 20)$ as $- 1 (14 n^{2} + 29 n + 20)$ .

$\frac{- 1 (14 n^{2} + 29 n + 20)}{n (n + 2)}$

Step 5.6.2

Move the negative in front of the fraction.

$- \frac{14 n^{2} + 29 n + 20}{n (n + 2)}$