Basic Math Examples

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

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

Step 1

Simplify each term.

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Step 1.1

Rewrite

{(a + b)}^{2}

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

(a + b) (a + b)

$(a + b) (a + b)$ .

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

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

Step 1.2

Expand

(a + b) (a + b)

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

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Step 1.2.1

Apply the distributive property.

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

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

Step 1.2.2

Apply the distributive property.

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

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

Step 1.2.3

Apply the distributive property.

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

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

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

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply

a

$a$ by

a

$a$ .

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

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

Step 1.3.1.2

Multiply

b

$b$ by

b

$b$ .

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

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

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

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

Step 1.3.2

Add

a b

$a b$ and

b a

$b a$ .

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Step 1.3.2.1

Reorder

b

$b$ and

a

$a$ .

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

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

Step 1.3.2.2

Add

a b

$a b$ and

a b

$a b$ .

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

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

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

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

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

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

Step 1.4

Rewrite

{(a - b)}^{2}

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

(a - b) (a - b)

$(a - b) (a - b)$ .

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

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

Step 1.5

Expand

(a - b) (a - b)

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

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Step 1.5.1

Apply the distributive property.

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

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

Step 1.5.2

Apply the distributive property.

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

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

Step 1.5.3

Apply the distributive property.

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

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

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

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

Step 1.6

Simplify and combine like terms.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

Multiply

a

$a$ by

a

$a$ .

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

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

Step 1.6.1.2

Rewrite using the commutative property of multiplication.

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

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

Step 1.6.1.3

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)

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

Step 1.6.1.4

Multiply

b

$b$ by

b

$b$ by adding the exponents.

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Step 1.6.1.4.1

Move

b

$b$ .

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

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

Step 1.6.1.4.2

Multiply

b

$b$ by

b

$b$ .

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

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

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

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

Step 1.6.1.5

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

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

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

Step 1.6.1.6

Multiply

b^{2}

$b^{2}$ by

1

$1$ .

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

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

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

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

Step 1.6.2

Subtract

b a

$b a$ from

- a b

$- a b$ .

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Step 1.6.2.1

Move

b

$b$ .

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

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

Step 1.6.2.2

Subtract

a b

$a b$ from

- a b

$- a b$ .

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

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

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

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

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

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

Step 1.7

Apply the distributive property.

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

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

Step 1.8

Multiply

- 2

$- 2$ by

- 1

$- 1$ .

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

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

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

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

Step 2

Simplify by adding terms.

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Step 2.1

Combine the opposite terms in

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

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

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Step 2.1.1

Subtract

a^{2}

$a^{2}$ from

a^{2}

$a^{2}$ .

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

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

Step 2.1.2

Add

2 a b + b^{2}

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

0

$0$ .

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

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

Step 2.1.3

Subtract

b^{2}

$b^{2}$ from

b^{2}

$b^{2}$ .

2 a b + 2 a b + 0

$2 a b + 2 a b + 0$

Step 2.1.4

Add

2 a b + 2 a b

$2 a b + 2 a b$ and

0

$0$ .

2 a b + 2 a b

$2 a b + 2 a b$

2 a b + 2 a b

$2 a b + 2 a b$

Step 2.2

Add

2 a b

$2 a b$ and

2 a b

$2 a b$ .

4 a b

$4 a b$

4 a b

$4 a b$