Algebra Examples

$36 {(m + n)}^{2} - 121 {(m - n)}^{2}$

Step 1

Rewrite $36 {(m + n)}^{2}$ as ${(6 (m + n))}^{2}$ .

${(6 (m + n))}^{2} - 121 {(m - n)}^{2}$

Step 2

Rewrite $121 {(m - n)}^{2}$ as ${(11 (m - n))}^{2}$ .

${(6 (m + n))}^{2} - {(11 (m - n))}^{2}$

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 6 (m + n)$ and $b = 11 (m - n)$ .

$(6 (m + n) + 11 (m - n)) (6 (m + n) - (11 (m - n)))$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$(6 m + 6 n + 11 (m - n)) (6 (m + n) - (11 (m - n)))$

Step 4.2

Apply the distributive property.

$(6 m + 6 n + 11 m + 11 (- n)) (6 (m + n) - (11 (m - n)))$

Step 4.3

Multiply $- 1$ by $11$ .

$(6 m + 6 n + 11 m - 11 n) (6 (m + n) - (11 (m - n)))$

Step 4.4

Add $6 m$ and $11 m$ .

$(17 m + 6 n - 11 n) (6 (m + n) - (11 (m - n)))$

Step 4.5

Subtract $11 n$ from $6 n$ .

$(17 m - 5 n) (6 (m + n) - (11 (m - n)))$

Step 4.6

Apply the distributive property.

$(17 m - 5 n) (6 m + 6 n - (11 (m - n)))$

Step 4.7

Apply the distributive property.

$(17 m - 5 n) (6 m + 6 n - (11 m + 11 (- n)))$

Step 4.8

Multiply $- 1$ by $11$ .

$(17 m - 5 n) (6 m + 6 n - (11 m - 11 n))$

Step 4.9

Apply the distributive property.

$(17 m - 5 n) (6 m + 6 n - (11 m) - (- 11 n))$

Step 4.10

Multiply $11$ by $- 1$ .

$(17 m - 5 n) (6 m + 6 n - 11 m - (- 11 n))$

Step 4.11

Multiply $- 11$ by $- 1$ .

$(17 m - 5 n) (6 m + 6 n - 11 m + 11 n)$

Step 4.12

Subtract $11 m$ from $6 m$ .

$(17 m - 5 n) (- 5 m + 6 n + 11 n)$

Step 4.13

Add $6 n$ and $11 n$ .

$(17 m - 5 n) (- 5 m + 17 n)$

Step 5

Factor.

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Step 5.1

Factor $- 1$ out of $- 5 m + 17 n$ .

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Step 5.1.1

Factor $- 1$ out of $- 5 m$ .

$(17 m - 5 n) (- (5 m) + 17 n)$

Step 5.1.2

Factor $- 1$ out of $17 n$ .

$(17 m - 5 n) (- (5 m) - (- 17 n))$

Step 5.1.3

Factor $- 1$ out of $- (5 m) - (- 17 n)$ .

$(17 m - 5 n) (- (5 m - 17 n))$

Step 5.2

Remove unnecessary parentheses.

$(17 m - 5 n) \cdot - 1 (5 m - 17 n)$

Step 6

Factor out negative.

$- ((17 m - 5 n) (5 m - 17 n))$

Step 7

Remove unnecessary parentheses.

$- (17 m - 5 n) (5 m - 17 n)$