Algebra Examples

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

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Simplify each term.

Tap for more steps...

Step 1.3.1

Multiply $a$ by $a$ .

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

Step 1.3.2

Multiply $b$ by $b$ .

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

Step 1.3.3

Multiply $c$ by $c$ .

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

Step 1.4

Add $a b$ and $b a$ .

Tap for more steps...

Step 1.4.1

Reorder $b$ and $a$ .

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

Step 1.4.2

Add $a b$ and $a b$ .

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

Step 1.5

Add $a c$ and $c a$ .

Tap for more steps...

Step 1.5.1

Reorder $c$ and $a$ .

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

Step 1.5.2

Add $a c$ and $a c$ .

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

Step 1.6

Add $b c$ and $c b$ .

Tap for more steps...

Step 1.6.1

Reorder $c$ and $b$ .

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

Step 1.6.2

Add $b c$ and $b c$ .

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

Step 1.7

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

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

Step 1.8

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

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

Step 1.9

Simplify each term.

Tap for more steps...

Step 1.9.1

Multiply $a$ by $a$ .

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

Step 1.9.2

Rewrite using the commutative property of multiplication.

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

Step 1.9.3

Rewrite using the commutative property of multiplication.

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

Step 1.9.4

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

Tap for more steps...

Step 1.9.4.1

Move $b$ .

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

Step 1.9.4.2

Multiply $b$ by $b$ .

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

Step 1.9.5

Multiply $- 1$ by $- 1$ .

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

Step 1.9.6

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

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

Step 1.9.7

Rewrite using the commutative property of multiplication.

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

Step 1.9.8

Multiply $c$ by $c$ .

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

Step 1.10

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

Tap for more steps...

Step 1.10.1

Move $b$ .

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

Step 1.10.2

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

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

Step 1.11

Add $a c$ and $c a$ .

Tap for more steps...

Step 1.11.1

Reorder $c$ and $a$ .

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

Step 1.11.2

Add $a c$ and $a c$ .

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

Step 1.12

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

Tap for more steps...

Step 1.12.1

Move $c$ .

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

Step 1.12.2

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

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

Step 1.13

Apply the distributive property.

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

Step 1.14

Simplify.

Tap for more steps...

Step 1.14.1

Multiply $- 2$ by $- 1$ .

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

Step 1.14.2

Multiply $- 2$ by $- 1$ .

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

Step 1.14.3

Multiply $2$ by $- 1$ .

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

Step 1.15

Remove parentheses.

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

Step 2

Simplify by adding terms.

Tap for more steps...

Step 2.1

Combine the opposite terms in $a^{2} + 2 a b + b^{2} + 2 b c + 2 a c + c^{2} - a^{2} + 2 a b - b^{2} + 2 b c - 2 a c - c^{2}$ .

Tap for more steps...

Step 2.1.1

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

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

Step 2.1.2

Add $2 a b + b^{2} + 2 b c + 2 a c + c^{2}$ and $0$ .

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

Step 2.1.3

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

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

Step 2.1.4

Add $2 a b + 2 b c + 2 a c + c^{2} + 2 a b$ and $0$ .

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

Step 2.1.5

Subtract $2 a c$ from $2 a c$ .

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

Step 2.1.6

Add $2 a b + 2 b c + c^{2} + 2 a b + 2 b c$ and $0$ .

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

Step 2.1.7

Subtract $c^{2}$ from $c^{2}$ .

$2 a b + 2 b c + 2 a b + 2 b c + 0$

Step 2.1.8

Add $2 a b + 2 b c + 2 a b + 2 b c$ and $0$ .

$2 a b + 2 b c + 2 a b + 2 b c$

Step 2.2

Add $2 a b$ and $2 a b$ .

$4 a b + 2 b c + 2 b c$

Step 2.3

Add $2 b c$ and $2 b c$ .

$4 a b + 4 b c$