Basic Math Examples

${(a - b)}^{2} + {(a - b)}^{2}$

Step 1

Rewrite ${(a - b)}^{2}$ as $(a - b) (a - b)$ .

$(a - b) (a - b) + {(a - b)}^{2}$

Step 2

Expand $(a - b) (a - b)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$a (a - b) - b (a - b) + {(a - b)}^{2}$

Step 2.2

Apply the distributive property.

$a \cdot a + a (- b) - b (a - b) + {(a - b)}^{2}$

Step 2.3

Apply the distributive property.

$a \cdot a + a (- b) - b a - b (- b) + {(a - b)}^{2}$

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 $a$ by $a$ .

$a^{2} + a (- b) - b a - b (- b) + {(a - b)}^{2}$

Step 3.1.2

Rewrite using the commutative property of multiplication.

$a^{2} - a b - b a - b (- b) + {(a - b)}^{2}$

Step 3.1.3

Rewrite using the commutative property of multiplication.

$a^{2} - a b - b a - 1 \cdot - 1 b \cdot b + {(a - b)}^{2}$

Step 3.1.4

Multiply $b$ by $b$ by adding the exponents.

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Step 3.1.4.1

Move $b$ .

$a^{2} - a b - b a - 1 \cdot - 1 (b \cdot b) + {(a - b)}^{2}$

Step 3.1.4.2

Multiply $b$ by $b$ .

$a^{2} - a b - b a - 1 \cdot - 1 b^{2} + {(a - b)}^{2}$

Step 3.1.5

Multiply $- 1$ by $- 1$ .

$a^{2} - a b - b a + 1 b^{2} + {(a - b)}^{2}$

Step 3.1.6

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

$a^{2} - a b - b a + b^{2} + {(a - b)}^{2}$

Step 3.2

Subtract $b a$ from $- a b$ .

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Step 3.2.1

Move $b$ .

$a^{2} - a b - 1 a b + b^{2} + {(a - b)}^{2}$

Step 3.2.2

Subtract $a b$ from $- a b$ .

$a^{2} - 2 a b + b^{2} + {(a - b)}^{2}$

Step 4

Rewrite ${(a - b)}^{2}$ as $(a - b) (a - b)$ .

$a^{2} - 2 a b + b^{2} + (a - b) (a - b)$

Step 5

Expand $(a - b) (a - b)$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$a^{2} - 2 a b + b^{2} + a (a - b) - b (a - b)$

Step 5.2

Apply the distributive property.

$a^{2} - 2 a b + b^{2} + a \cdot a + a (- b) - b (a - b)$

Step 5.3

Apply the distributive property.

$a^{2} - 2 a b + b^{2} + a \cdot a + a (- b) - b a - b (- b)$

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 $a$ by $a$ .

$a^{2} - 2 a b + b^{2} + a^{2} + a (- b) - b a - b (- b)$

Step 6.1.2

Rewrite using the commutative property of multiplication.

$a^{2} - 2 a b + b^{2} + a^{2} - a b - b a - b (- b)$

Step 6.1.3

Rewrite using the commutative property of multiplication.

$a^{2} - 2 a b + b^{2} + a^{2} - a b - b a - 1 \cdot - 1 b \cdot b$

Step 6.1.4

Multiply $b$ by $b$ by adding the exponents.

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Step 6.1.4.1

Move $b$ .

$a^{2} - 2 a b + b^{2} + a^{2} - a b - b a - 1 \cdot - 1 (b \cdot b)$

Step 6.1.4.2

Multiply $b$ by $b$ .

$a^{2} - 2 a b + b^{2} + a^{2} - a b - b a - 1 \cdot - 1 b^{2}$

Step 6.1.5

Multiply $- 1$ by $- 1$ .

$a^{2} - 2 a b + b^{2} + a^{2} - a b - b a + 1 b^{2}$

Step 6.1.6

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

$a^{2} - 2 a b + b^{2} + a^{2} - a b - b a + b^{2}$

Step 6.2

Subtract $b a$ from $- a b$ .

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Step 6.2.1

Move $b$ .

$a^{2} - 2 a b + b^{2} + a^{2} - a b - 1 a b + b^{2}$

Step 6.2.2

Subtract $a b$ from $- a b$ .

$a^{2} - 2 a b + b^{2} + a^{2} - 2 a b + b^{2}$

Step 7

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

$2 a^{2} - 2 a b + b^{2} - 2 a b + b^{2}$

Step 8

Subtract $2 a b$ from $- 2 a b$ .

$2 a^{2} + b^{2} - 4 a b + b^{2}$

Step 9

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

$2 a^{2} + 2 b^{2} - 4 a b$

Step 10

Rewrite $2 a^{2} + 2 b^{2} - 4 a b$ in a factored form.

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Step 10.1

Factor $2$ out of $2 a^{2} + 2 b^{2} - 4 a b$ .

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Step 10.1.1

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

$2 (a^{2}) + 2 b^{2} - 4 a b$

Step 10.1.2

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

$2 (a^{2}) + 2 (b^{2}) - 4 a b$

Step 10.1.3

Factor $2$ out of $- 4 a b$ .

$2 (a^{2}) + 2 (b^{2}) + 2 (- 2 a b)$

Step 10.1.4

Factor $2$ out of $2 (a^{2}) + 2 (b^{2})$ .

$2 (a^{2} + b^{2}) + 2 (- 2 a b)$

Step 10.1.5

Factor $2$ out of $2 (a^{2} + b^{2}) + 2 (- 2 a b)$ .

$2 (a^{2} + b^{2} - 2 a b)$

Step 10.2

Factor using the perfect square rule.

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Step 10.2.1

Rearrange terms.

$2 (a^{2} - 2 a b + b^{2})$

Step 10.2.2

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

$2 a b = 2 \cdot a \cdot b$

Step 10.2.3

Rewrite the polynomial.

$2 (a^{2} - 2 \cdot a \cdot b + b^{2})$

Step 10.2.4

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

$2 {(a - b)}^{2}$