Basic Math Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Combine the opposite terms in $a \cdot a + a (- b) + b a + b (- b)$ .

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Step 2.1

Reorder the factors in the terms $a (- b)$ and $b a$ .

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

Step 2.2

Add $- a b$ and $a b$ .

$a \cdot a + 0 + b (- b) + {(a + b)}^{2} - 2 a b$

Step 2.3

Add $a \cdot a$ and $0$ .

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

Step 3

Simplify each term.

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Step 3.1

Multiply $a$ by $a$ .

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

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

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Step 3.3.1

Move $b$ .

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

Step 3.3.2

Multiply $b$ by $b$ .

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

Step 4

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

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

Step 5

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

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

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 $a$ .

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

Step 6.1.2

Multiply $b$ by $b$ .

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

Step 6.2

Add $a b$ and $b a$ .

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Step 6.2.1

Reorder $b$ and $a$ .

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

Step 6.2.2

Add $a b$ and $a b$ .

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

Step 7

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

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

Step 8

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

$2 a^{2} + 2 a b + 0 - 2 a b$

Step 9

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

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

Step 10

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

$2 a^{2} + 0$

Step 11

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

$2 a^{2}$