Finite Math Examples

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

Step 1

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

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

Step 2

Expand $(m + n) (m + n)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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 $m$ by $m$ .

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

Step 3.1.2

Multiply $n$ by $n$ .

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

Step 3.2

Add $m n$ and $n m$ .

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Step 3.2.1

Reorder $n$ and $m$ .

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

Step 3.2.2

Add $m n$ and $m n$ .

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

Step 4

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

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

Step 5

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

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $m$ by $m$ .

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

Step 6.1.2

Rewrite using the commutative property of multiplication.

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

Step 6.1.3

Rewrite using the commutative property of multiplication.

$m^{2} + 2 m n + n^{2} + m^{2} - m n - n m - 1 \cdot - 1 n \cdot n$

Step 6.1.4

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

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Step 6.1.4.1

Move $n$ .

$m^{2} + 2 m n + n^{2} + m^{2} - m n - n m - 1 \cdot - 1 (n \cdot n)$

Step 6.1.4.2

Multiply $n$ by $n$ .

$m^{2} + 2 m n + n^{2} + m^{2} - m n - n m - 1 \cdot - 1 n^{2}$

Step 6.1.5

Multiply $- 1$ by $- 1$ .

$m^{2} + 2 m n + n^{2} + m^{2} - m n - n m + 1 n^{2}$

Step 6.1.6

Multiply $n^{2}$ by $1$ .

$m^{2} + 2 m n + n^{2} + m^{2} - m n - n m + n^{2}$

Step 6.2

Subtract $n m$ from $- m n$ .

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Step 6.2.1

Move $n$ .

$m^{2} + 2 m n + n^{2} + m^{2} - m n - 1 m n + n^{2}$

Step 6.2.2

Subtract $m n$ from $- m n$ .

$m^{2} + 2 m n + n^{2} + m^{2} - 2 m n + n^{2}$

Step 7

Add $m^{2}$ and $m^{2}$ .

$2 m^{2} + 2 m n + n^{2} - 2 m n + n^{2}$

Step 8

Subtract $2 m n$ from $2 m n$ .

$2 m^{2} + n^{2} + 0 + n^{2}$

Step 9

Add $2 m^{2}$ and $0$ .

$2 m^{2} + n^{2} + n^{2}$

Step 10

Add $n^{2}$ and $n^{2}$ .

$2 m^{2} + 2 n^{2}$

Step 11

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

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Step 11.1

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

$2 (m^{2}) + 2 n^{2}$

Step 11.2

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

$2 (m^{2}) + 2 (n^{2})$

Step 11.3

Factor $2$ out of $2 (m^{2}) + 2 (n^{2})$ .

$2 (m^{2} + n^{2})$