Algebra Examples

${(3 a + 2 b)}^{2} - {(a + 3 b + 2 c)}^{2}$

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(3 a + 2 b)}^{2}$ as $(3 a + 2 b) (3 a + 2 b)$ .

$(3 a + 2 b) (3 a + 2 b) - {(a + 3 b + 2 c)}^{2}$

Step 1.2

Expand $(3 a + 2 b) (3 a + 2 b)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$3 a (3 a + 2 b) + 2 b (3 a + 2 b) - {(a + 3 b + 2 c)}^{2}$

Step 1.2.2

Apply the distributive property.

$3 a (3 a) + 3 a (2 b) + 2 b (3 a + 2 b) - {(a + 3 b + 2 c)}^{2}$

Step 1.2.3

Apply the distributive property.

$3 a (3 a) + 3 a (2 b) + 2 b (3 a) + 2 b (2 b) - {(a + 3 b + 2 c)}^{2}$

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

Rewrite using the commutative property of multiplication.

$3 \cdot 3 a \cdot a + 3 a (2 b) + 2 b (3 a) + 2 b (2 b) - {(a + 3 b + 2 c)}^{2}$

Step 1.3.1.2

Multiply $a$ by $a$ by adding the exponents.

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Step 1.3.1.2.1

Move $a$ .

$3 \cdot 3 (a \cdot a) + 3 a (2 b) + 2 b (3 a) + 2 b (2 b) - {(a + 3 b + 2 c)}^{2}$

Step 1.3.1.2.2

Multiply $a$ by $a$ .

$3 \cdot 3 a^{2} + 3 a (2 b) + 2 b (3 a) + 2 b (2 b) - {(a + 3 b + 2 c)}^{2}$

Step 1.3.1.3

Multiply $3$ by $3$ .

$9 a^{2} + 3 a (2 b) + 2 b (3 a) + 2 b (2 b) - {(a + 3 b + 2 c)}^{2}$

Step 1.3.1.4

Rewrite using the commutative property of multiplication.

$9 a^{2} + 3 \cdot 2 a b + 2 b (3 a) + 2 b (2 b) - {(a + 3 b + 2 c)}^{2}$

Step 1.3.1.5

Multiply $3$ by $2$ .

$9 a^{2} + 6 a b + 2 b (3 a) + 2 b (2 b) - {(a + 3 b + 2 c)}^{2}$

Step 1.3.1.6

Rewrite using the commutative property of multiplication.

$9 a^{2} + 6 a b + 2 \cdot 3 b a + 2 b (2 b) - {(a + 3 b + 2 c)}^{2}$

Step 1.3.1.7

Multiply $2$ by $3$ .

$9 a^{2} + 6 a b + 6 b a + 2 b (2 b) - {(a + 3 b + 2 c)}^{2}$

Step 1.3.1.8

Rewrite using the commutative property of multiplication.

$9 a^{2} + 6 a b + 6 b a + 2 \cdot 2 b \cdot b - {(a + 3 b + 2 c)}^{2}$

Step 1.3.1.9

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

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Step 1.3.1.9.1

Move $b$ .

$9 a^{2} + 6 a b + 6 b a + 2 \cdot 2 (b \cdot b) - {(a + 3 b + 2 c)}^{2}$

Step 1.3.1.9.2

Multiply $b$ by $b$ .

$9 a^{2} + 6 a b + 6 b a + 2 \cdot 2 b^{2} - {(a + 3 b + 2 c)}^{2}$

Step 1.3.1.10

Multiply $2$ by $2$ .

$9 a^{2} + 6 a b + 6 b a + 4 b^{2} - {(a + 3 b + 2 c)}^{2}$

Step 1.3.2

Add $6 a b$ and $6 b a$ .

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Step 1.3.2.1

Move $b$ .

$9 a^{2} + 6 a b + 6 a b + 4 b^{2} - {(a + 3 b + 2 c)}^{2}$

Step 1.3.2.2

Add $6 a b$ and $6 a b$ .

$9 a^{2} + 12 a b + 4 b^{2} - {(a + 3 b + 2 c)}^{2}$

Step 1.4

Rewrite ${(a + 3 b + 2 c)}^{2}$ as $(a + 3 b + 2 c) (a + 3 b + 2 c)$ .

$9 a^{2} + 12 a b + 4 b^{2} - ((a + 3 b + 2 c) (a + 3 b + 2 c))$

Step 1.5

Expand $(a + 3 b + 2 c) (a + 3 b + 2 c)$ by multiplying each term in the first expression by each term in the second expression.

$9 a^{2} + 12 a b + 4 b^{2} - (a \cdot a + a (3 b) + a (2 c) + 3 b a + 3 b (3 b) + 3 b (2 c) + 2 c a + 2 c (3 b) + 2 c (2 c))$

Step 1.6

Simplify each term.

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Step 1.6.1

Multiply $a$ by $a$ .

$9 a^{2} + 12 a b + 4 b^{2} - (a^{2} + a (3 b) + a (2 c) + 3 b a + 3 b (3 b) + 3 b (2 c) + 2 c a + 2 c (3 b) + 2 c (2 c))$

Step 1.6.2

Rewrite using the commutative property of multiplication.

$9 a^{2} + 12 a b + 4 b^{2} - (a^{2} + 3 a b + a (2 c) + 3 b a + 3 b (3 b) + 3 b (2 c) + 2 c a + 2 c (3 b) + 2 c (2 c))$

Step 1.6.3

Rewrite using the commutative property of multiplication.

$9 a^{2} + 12 a b + 4 b^{2} - (a^{2} + 3 a b + 2 a c + 3 b a + 3 b (3 b) + 3 b (2 c) + 2 c a + 2 c (3 b) + 2 c (2 c))$

Step 1.6.4

Rewrite using the commutative property of multiplication.

$9 a^{2} + 12 a b + 4 b^{2} - (a^{2} + 3 a b + 2 a c + 3 b a + 3 \cdot 3 b \cdot b + 3 b (2 c) + 2 c a + 2 c (3 b) + 2 c (2 c))$

Step 1.6.5

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

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Step 1.6.5.1

Move $b$ .

$9 a^{2} + 12 a b + 4 b^{2} - (a^{2} + 3 a b + 2 a c + 3 b a + 3 \cdot 3 (b \cdot b) + 3 b (2 c) + 2 c a + 2 c (3 b) + 2 c (2 c))$

Step 1.6.5.2

Multiply $b$ by $b$ .

$9 a^{2} + 12 a b + 4 b^{2} - (a^{2} + 3 a b + 2 a c + 3 b a + 3 \cdot 3 b^{2} + 3 b (2 c) + 2 c a + 2 c (3 b) + 2 c (2 c))$

Step 1.6.6

Multiply $3$ by $3$ .

$9 a^{2} + 12 a b + 4 b^{2} - (a^{2} + 3 a b + 2 a c + 3 b a + 9 b^{2} + 3 b (2 c) + 2 c a + 2 c (3 b) + 2 c (2 c))$

Step 1.6.7

Rewrite using the commutative property of multiplication.

$9 a^{2} + 12 a b + 4 b^{2} - (a^{2} + 3 a b + 2 a c + 3 b a + 9 b^{2} + 3 \cdot 2 b c + 2 c a + 2 c (3 b) + 2 c (2 c))$

Step 1.6.8

Multiply $3$ by $2$ .

$9 a^{2} + 12 a b + 4 b^{2} - (a^{2} + 3 a b + 2 a c + 3 b a + 9 b^{2} + 6 b c + 2 c a + 2 c (3 b) + 2 c (2 c))$

Step 1.6.9

Rewrite using the commutative property of multiplication.

$9 a^{2} + 12 a b + 4 b^{2} - (a^{2} + 3 a b + 2 a c + 3 b a + 9 b^{2} + 6 b c + 2 c a + 2 \cdot 3 c b + 2 c (2 c))$

Step 1.6.10

Multiply $2$ by $3$ .

$9 a^{2} + 12 a b + 4 b^{2} - (a^{2} + 3 a b + 2 a c + 3 b a + 9 b^{2} + 6 b c + 2 c a + 6 c b + 2 c (2 c))$

Step 1.6.11

Rewrite using the commutative property of multiplication.

$9 a^{2} + 12 a b + 4 b^{2} - (a^{2} + 3 a b + 2 a c + 3 b a + 9 b^{2} + 6 b c + 2 c a + 6 c b + 2 \cdot 2 c \cdot c)$

Step 1.6.12

Multiply $c$ by $c$ by adding the exponents.

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Step 1.6.12.1

Move $c$ .

$9 a^{2} + 12 a b + 4 b^{2} - (a^{2} + 3 a b + 2 a c + 3 b a + 9 b^{2} + 6 b c + 2 c a + 6 c b + 2 \cdot 2 (c \cdot c))$

Step 1.6.12.2

Multiply $c$ by $c$ .

$9 a^{2} + 12 a b + 4 b^{2} - (a^{2} + 3 a b + 2 a c + 3 b a + 9 b^{2} + 6 b c + 2 c a + 6 c b + 2 \cdot 2 c^{2})$

Step 1.6.13

Multiply $2$ by $2$ .

$9 a^{2} + 12 a b + 4 b^{2} - (a^{2} + 3 a b + 2 a c + 3 b a + 9 b^{2} + 6 b c + 2 c a + 6 c b + 4 c^{2})$

Step 1.7

Add $3 a b$ and $3 b a$ .

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Step 1.7.1

Move $b$ .

$9 a^{2} + 12 a b + 4 b^{2} - (a^{2} + 3 a b + 3 a b + 2 a c + 9 b^{2} + 6 b c + 2 c a + 6 c b + 4 c^{2})$

Step 1.7.2

Add $3 a b$ and $3 a b$ .

$9 a^{2} + 12 a b + 4 b^{2} - (a^{2} + 6 a b + 2 a c + 9 b^{2} + 6 b c + 2 c a + 6 c b + 4 c^{2})$

Step 1.8

Add $2 a c$ and $2 c a$ .

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Step 1.8.1

Move $c$ .

$9 a^{2} + 12 a b + 4 b^{2} - (a^{2} + 6 a b + 9 b^{2} + 6 b c + 2 a c + 2 a c + 6 c b + 4 c^{2})$

Step 1.8.2

Add $2 a c$ and $2 a c$ .

$9 a^{2} + 12 a b + 4 b^{2} - (a^{2} + 6 a b + 9 b^{2} + 6 b c + 4 a c + 6 c b + 4 c^{2})$

Step 1.9

Add $6 b c$ and $6 c b$ .

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Step 1.9.1

Move $c$ .

$9 a^{2} + 12 a b + 4 b^{2} - (a^{2} + 6 a b + 9 b^{2} + 6 b c + 6 b c + 4 a c + 4 c^{2})$

Step 1.9.2

Add $6 b c$ and $6 b c$ .

$9 a^{2} + 12 a b + 4 b^{2} - (a^{2} + 6 a b + 9 b^{2} + 12 b c + 4 a c + 4 c^{2})$

Step 1.10

Apply the distributive property.

$9 a^{2} + 12 a b + 4 b^{2} - a^{2} - (6 a b) - (9 b^{2}) - (12 b c) - (4 a c) - (4 c^{2})$

Step 1.11

Simplify.

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Step 1.11.1

Multiply $6$ by $- 1$ .

$9 a^{2} + 12 a b + 4 b^{2} - a^{2} - 6 (a b) - (9 b^{2}) - (12 b c) - (4 a c) - (4 c^{2})$

Step 1.11.2

Multiply $9$ by $- 1$ .

$9 a^{2} + 12 a b + 4 b^{2} - a^{2} - 6 (a b) - 9 b^{2} - (12 b c) - (4 a c) - (4 c^{2})$

Step 1.11.3

Multiply $12$ by $- 1$ .

$9 a^{2} + 12 a b + 4 b^{2} - a^{2} - 6 (a b) - 9 b^{2} - 12 (b c) - (4 a c) - (4 c^{2})$

Step 1.11.4

Multiply $4$ by $- 1$ .

$9 a^{2} + 12 a b + 4 b^{2} - a^{2} - 6 (a b) - 9 b^{2} - 12 (b c) - 4 (a c) - (4 c^{2})$

Step 1.11.5

Multiply $4$ by $- 1$ .

$9 a^{2} + 12 a b + 4 b^{2} - a^{2} - 6 (a b) - 9 b^{2} - 12 (b c) - 4 (a c) - 4 c^{2}$

Step 1.12

Remove parentheses.

$9 a^{2} + 12 a b + 4 b^{2} - a^{2} - 6 a b - 9 b^{2} - 12 b c - 4 a c - 4 c^{2}$

Step 2

Simplify by adding terms.

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Step 2.1

Subtract $a^{2}$ from $9 a^{2}$ .

$8 a^{2} + 12 a b + 4 b^{2} - 6 a b - 9 b^{2} - 12 b c - 4 a c - 4 c^{2}$

Step 2.2

Subtract $6 a b$ from $12 a b$ .

$8 a^{2} + 4 b^{2} + 6 a b - 9 b^{2} - 12 b c - 4 a c - 4 c^{2}$

Step 2.3

Subtract $9 b^{2}$ from $4 b^{2}$ .

$8 a^{2} - 5 b^{2} + 6 a b - 12 b c - 4 a c - 4 c^{2}$

Step 2.4

Reorder.

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Step 2.4.1

Move $- 12 b c$ .

$8 a^{2} - 5 b^{2} + 6 a b - 4 a c - 12 b c - 4 c^{2}$

Step 2.4.2

Move $- 5 b^{2}$ .

$8 a^{2} + 6 a b - 4 a c - 5 b^{2} - 12 b c - 4 c^{2}$