Basic Math Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify each term.

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Step 3.1

Rewrite using the commutative property of multiplication.

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

Step 3.2

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

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Step 3.2.1

Move $a$ .

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

Step 3.2.2

Multiply $a$ by $a$ .

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

Step 3.3

Multiply $4$ by $4$ .

$16 a^{2} + 4 a (- b) + 4 a (2 c) - b (4 a) - b (- b) - b (2 c) + 2 c (4 a) + 2 c (- b) + 2 c (2 c)$

Step 3.4

Rewrite using the commutative property of multiplication.

$16 a^{2} + 4 \cdot - 1 a b + 4 a (2 c) - b (4 a) - b (- b) - b (2 c) + 2 c (4 a) + 2 c (- b) + 2 c (2 c)$

Step 3.5

Multiply $4$ by $- 1$ .

$16 a^{2} - 4 a b + 4 a (2 c) - b (4 a) - b (- b) - b (2 c) + 2 c (4 a) + 2 c (- b) + 2 c (2 c)$

Step 3.6

Rewrite using the commutative property of multiplication.

$16 a^{2} - 4 a b + 4 \cdot 2 a c - b (4 a) - b (- b) - b (2 c) + 2 c (4 a) + 2 c (- b) + 2 c (2 c)$

Step 3.7

Multiply $4$ by $2$ .

$16 a^{2} - 4 a b + 8 a c - b (4 a) - b (- b) - b (2 c) + 2 c (4 a) + 2 c (- b) + 2 c (2 c)$

Step 3.8

Rewrite using the commutative property of multiplication.

$16 a^{2} - 4 a b + 8 a c - 1 \cdot 4 b a - b (- b) - b (2 c) + 2 c (4 a) + 2 c (- b) + 2 c (2 c)$

Step 3.9

Multiply $- 1$ by $4$ .

$16 a^{2} - 4 a b + 8 a c - 4 b a - b (- b) - b (2 c) + 2 c (4 a) + 2 c (- b) + 2 c (2 c)$

Step 3.10

Rewrite using the commutative property of multiplication.

$16 a^{2} - 4 a b + 8 a c - 4 b a - 1 \cdot - 1 b \cdot b - b (2 c) + 2 c (4 a) + 2 c (- b) + 2 c (2 c)$

Step 3.11

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

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Step 3.11.1

Move $b$ .

$16 a^{2} - 4 a b + 8 a c - 4 b a - 1 \cdot - 1 (b \cdot b) - b (2 c) + 2 c (4 a) + 2 c (- b) + 2 c (2 c)$

Step 3.11.2

Multiply $b$ by $b$ .

$16 a^{2} - 4 a b + 8 a c - 4 b a - 1 \cdot - 1 b^{2} - b (2 c) + 2 c (4 a) + 2 c (- b) + 2 c (2 c)$

Step 3.12

Multiply $- 1$ by $- 1$ .

$16 a^{2} - 4 a b + 8 a c - 4 b a + 1 b^{2} - b (2 c) + 2 c (4 a) + 2 c (- b) + 2 c (2 c)$

Step 3.13

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

$16 a^{2} - 4 a b + 8 a c - 4 b a + b^{2} - b (2 c) + 2 c (4 a) + 2 c (- b) + 2 c (2 c)$

Step 3.14

Rewrite using the commutative property of multiplication.

$16 a^{2} - 4 a b + 8 a c - 4 b a + b^{2} - 1 \cdot 2 b c + 2 c (4 a) + 2 c (- b) + 2 c (2 c)$

Step 3.15

Multiply $- 1$ by $2$ .

$16 a^{2} - 4 a b + 8 a c - 4 b a + b^{2} - 2 b c + 2 c (4 a) + 2 c (- b) + 2 c (2 c)$

Step 3.16

Rewrite using the commutative property of multiplication.

$16 a^{2} - 4 a b + 8 a c - 4 b a + b^{2} - 2 b c + 2 \cdot 4 c a + 2 c (- b) + 2 c (2 c)$

Step 3.17

Multiply $2$ by $4$ .

$16 a^{2} - 4 a b + 8 a c - 4 b a + b^{2} - 2 b c + 8 c a + 2 c (- b) + 2 c (2 c)$

Step 3.18

Rewrite using the commutative property of multiplication.

$16 a^{2} - 4 a b + 8 a c - 4 b a + b^{2} - 2 b c + 8 c a + 2 \cdot - 1 c b + 2 c (2 c)$

Step 3.19

Multiply $2$ by $- 1$ .

$16 a^{2} - 4 a b + 8 a c - 4 b a + b^{2} - 2 b c + 8 c a - 2 c b + 2 c (2 c)$

Step 3.20

Rewrite using the commutative property of multiplication.

$16 a^{2} - 4 a b + 8 a c - 4 b a + b^{2} - 2 b c + 8 c a - 2 c b + 2 \cdot 2 c \cdot c$

Step 3.21

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

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Step 3.21.1

Move $c$ .

$16 a^{2} - 4 a b + 8 a c - 4 b a + b^{2} - 2 b c + 8 c a - 2 c b + 2 \cdot 2 (c \cdot c)$

Step 3.21.2

Multiply $c$ by $c$ .

$16 a^{2} - 4 a b + 8 a c - 4 b a + b^{2} - 2 b c + 8 c a - 2 c b + 2 \cdot 2 c^{2}$

Step 3.22

Multiply $2$ by $2$ .

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

Step 4

Subtract $4 b a$ from $- 4 a b$ .

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Step 4.1

Move $b$ .

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

Step 4.2

Subtract $4 a b$ from $- 4 a b$ .

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

Step 5

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

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Step 5.1

Move $c$ .

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

Step 5.2

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

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

Step 6

Subtract $2 c b$ from $- 2 b c$ .

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Step 6.1

Move $c$ .

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

Step 6.2

Subtract $2 b c$ from $- 2 b c$ .

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

Step 7

Move $- 4 b c$ .

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

Step 8

Move $b^{2}$ .

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