Basic Math Examples

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

Step 1

Rewrite ${(2 a + 3 b + 4 c)}^{2}$ as $(2 a + 3 b + 4 c) (2 a + 3 b + 4 c)$ .

$(2 a + 3 b + 4 c) (2 a + 3 b + 4 c)$

Step 2

Expand $(2 a + 3 b + 4 c) (2 a + 3 b + 4 c)$ by multiplying each term in the first expression by each term in the second expression.

$2 a (2 a) + 2 a (3 b) + 2 a (4 c) + 3 b (2 a) + 3 b (3 b) + 3 b (4 c) + 4 c (2 a) + 4 c (3 b) + 4 c (4 c)$

Step 3

Simplify each term.

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Step 3.1

Rewrite using the commutative property of multiplication.

$2 \cdot 2 a \cdot a + 2 a (3 b) + 2 a (4 c) + 3 b (2 a) + 3 b (3 b) + 3 b (4 c) + 4 c (2 a) + 4 c (3 b) + 4 c (4 c)$

Step 3.2

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

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Step 3.2.1

Move $a$ .

$2 \cdot 2 (a \cdot a) + 2 a (3 b) + 2 a (4 c) + 3 b (2 a) + 3 b (3 b) + 3 b (4 c) + 4 c (2 a) + 4 c (3 b) + 4 c (4 c)$

Step 3.2.2

Multiply $a$ by $a$ .

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

Step 3.3

Multiply $2$ by $2$ .

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

Step 3.4

Rewrite using the commutative property of multiplication.

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

Step 3.5

Multiply $2$ by $3$ .

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

Step 3.6

Rewrite using the commutative property of multiplication.

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

Step 3.7

Multiply $2$ by $4$ .

$4 a^{2} + 6 a b + 8 a c + 3 b (2 a) + 3 b (3 b) + 3 b (4 c) + 4 c (2 a) + 4 c (3 b) + 4 c (4 c)$

Step 3.8

Rewrite using the commutative property of multiplication.

$4 a^{2} + 6 a b + 8 a c + 3 \cdot 2 b a + 3 b (3 b) + 3 b (4 c) + 4 c (2 a) + 4 c (3 b) + 4 c (4 c)$

Step 3.9

Multiply $3$ by $2$ .

$4 a^{2} + 6 a b + 8 a c + 6 b a + 3 b (3 b) + 3 b (4 c) + 4 c (2 a) + 4 c (3 b) + 4 c (4 c)$

Step 3.10

Rewrite using the commutative property of multiplication.

$4 a^{2} + 6 a b + 8 a c + 6 b a + 3 \cdot 3 b \cdot b + 3 b (4 c) + 4 c (2 a) + 4 c (3 b) + 4 c (4 c)$

Step 3.11

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

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Step 3.11.1

Move $b$ .

$4 a^{2} + 6 a b + 8 a c + 6 b a + 3 \cdot 3 (b \cdot b) + 3 b (4 c) + 4 c (2 a) + 4 c (3 b) + 4 c (4 c)$

Step 3.11.2

Multiply $b$ by $b$ .

$4 a^{2} + 6 a b + 8 a c + 6 b a + 3 \cdot 3 b^{2} + 3 b (4 c) + 4 c (2 a) + 4 c (3 b) + 4 c (4 c)$

Step 3.12

Multiply $3$ by $3$ .

$4 a^{2} + 6 a b + 8 a c + 6 b a + 9 b^{2} + 3 b (4 c) + 4 c (2 a) + 4 c (3 b) + 4 c (4 c)$

Step 3.13

Rewrite using the commutative property of multiplication.

$4 a^{2} + 6 a b + 8 a c + 6 b a + 9 b^{2} + 3 \cdot 4 b c + 4 c (2 a) + 4 c (3 b) + 4 c (4 c)$

Step 3.14

Multiply $3$ by $4$ .

$4 a^{2} + 6 a b + 8 a c + 6 b a + 9 b^{2} + 12 b c + 4 c (2 a) + 4 c (3 b) + 4 c (4 c)$

Step 3.15

Rewrite using the commutative property of multiplication.

$4 a^{2} + 6 a b + 8 a c + 6 b a + 9 b^{2} + 12 b c + 4 \cdot 2 c a + 4 c (3 b) + 4 c (4 c)$

Step 3.16

Multiply $4$ by $2$ .

$4 a^{2} + 6 a b + 8 a c + 6 b a + 9 b^{2} + 12 b c + 8 c a + 4 c (3 b) + 4 c (4 c)$

Step 3.17

Rewrite using the commutative property of multiplication.

$4 a^{2} + 6 a b + 8 a c + 6 b a + 9 b^{2} + 12 b c + 8 c a + 4 \cdot 3 c b + 4 c (4 c)$

Step 3.18

Multiply $4$ by $3$ .

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

Step 3.19

Rewrite using the commutative property of multiplication.

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

Step 3.20

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

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Step 3.20.1

Move $c$ .

$4 a^{2} + 6 a b + 8 a c + 6 b a + 9 b^{2} + 12 b c + 8 c a + 12 c b + 4 \cdot 4 (c \cdot c)$

Step 3.20.2

Multiply $c$ by $c$ .

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

Step 3.21

Multiply $4$ by $4$ .

$4 a^{2} + 6 a b + 8 a c + 6 b a + 9 b^{2} + 12 b c + 8 c a + 12 c b + 16 c^{2}$

Step 4

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

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Step 4.1

Move $b$ .

$4 a^{2} + 6 a b + 6 a b + 8 a c + 9 b^{2} + 12 b c + 8 c a + 12 c b + 16 c^{2}$

Step 4.2

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

$4 a^{2} + 12 a b + 8 a c + 9 b^{2} + 12 b c + 8 c a + 12 c b + 16 c^{2}$

Step 5

Add $8 a c$ and $8 c a$ .

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Step 5.1

Move $c$ .

$4 a^{2} + 12 a b + 9 b^{2} + 12 b c + 8 a c + 8 a c + 12 c b + 16 c^{2}$

Step 5.2

Add $8 a c$ and $8 a c$ .

$4 a^{2} + 12 a b + 9 b^{2} + 12 b c + 16 a c + 12 c b + 16 c^{2}$

Step 6

Add $12 b c$ and $12 c b$ .

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Step 6.1

Move $c$ .

$4 a^{2} + 12 a b + 9 b^{2} + 12 b c + 12 b c + 16 a c + 16 c^{2}$

Step 6.2

Add $12 b c$ and $12 b c$ .

$4 a^{2} + 12 a b + 9 b^{2} + 24 b c + 16 a c + 16 c^{2}$

Step 7

Move $24 b c$ .

$4 a^{2} + 12 a b + 9 b^{2} + 16 a c + 24 b c + 16 c^{2}$

Step 8

Move $9 b^{2}$ .

$4 a^{2} + 12 a b + 16 a c + 9 b^{2} + 24 b c + 16 c^{2}$