Finite Math Examples

$2 {(n - 7)}^{2}$

Step 1

Rewrite ${(n - 7)}^{2}$ as $(n - 7) (n - 7)$ .

$2 ((n - 7) (n - 7))$

Step 2

Expand $(n - 7) (n - 7)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$2 (n (n - 7) - 7 (n - 7))$

Step 2.2

Apply the distributive property.

$2 (n \cdot n + n \cdot - 7 - 7 (n - 7))$

Step 2.3

Apply the distributive property.

$2 (n \cdot n + n \cdot - 7 - 7 n - 7 \cdot - 7)$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $n$ by $n$ .

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

Step 3.1.2

Move $- 7$ to the left of $n$ .

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

Step 3.1.3

Multiply $- 7$ by $- 7$ .

$2 (n^{2} - 7 n - 7 n + 49)$

Step 3.2

Subtract $7 n$ from $- 7 n$ .

$2 (n^{2} - 14 n + 49)$

Step 4

Apply the distributive property.

$2 n^{2} + 2 (- 14 n) + 2 \cdot 49$

Step 5

Simplify.

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Step 5.1

Multiply $- 14$ by $2$ .

$2 n^{2} - 28 n + 2 \cdot 49$

Step 5.2

Multiply $2$ by $49$ .

$2 n^{2} - 28 n + 98$

Step 6

Factor out the GCF of $2$ from $2 n^{2} - 28 n + 98$ .

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Step 6.1

Factor out the GCF of $2$ from each term in the polynomial.

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Step 6.1.1

Factor out the GCF of $2$ from the expression $2 n^{2}$ .

$2 (n^{2}) - 28 n + 98$

Step 6.1.2

Factor out the GCF of $2$ from the expression $- 28 n$ .

$2 (n^{2}) + 2 (- 14 n) + 98$

Step 6.1.3

Factor out the GCF of $2$ from the expression $98$ .

$2 (n^{2}) + 2 (- 14 n) + 2 (49)$

Step 6.2

Since all the terms share a common factor of $2$ , it can be factored out of each term.

$2 (n^{2} - 14 n + 49)$

Step 7

Factor using the perfect square rule.

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Step 7.1

Rewrite $49$ as $7^{2}$ .

$2 (n^{2} - 14 n + 7^{2})$

Step 7.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$14 n = 2 \cdot n \cdot 7$

Step 7.3

Rewrite the polynomial.

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

Step 7.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = n$ and $b = 7$ .

$2 ({(n - 7)}^{2})$