Algebra Examples

$(a + b - c) (a - b + C)$

Step 1

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

$a \cdot a + a (- b) + a C + b a + b (- b) + b C - c a - c (- b) - c C$

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $a \cdot a + a (- b) + a C + b a + b (- b) + b C - c a - c (- b) - c C$ .

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Step 2.1.1

Reorder the factors in the terms $a (- b)$ and $b a$ .

$a \cdot a - a b + a C + a b + b (- b) + b C - c a - c (- b) - c C$

Step 2.1.2

Add $- a b$ and $a b$ .

$a \cdot a + a C + 0 + b (- b) + b C - c a - c (- b) - c C$

Step 2.1.3

Add $a \cdot a + a C$ and $0$ .

$a \cdot a + a C + b (- b) + b C - c a - c (- b) - c C$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $a$ by $a$ .

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

Step 2.2.2

Rewrite using the commutative property of multiplication.

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

Step 2.2.3

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

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Step 2.2.3.1

Move $b$ .

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

Step 2.2.3.2

Multiply $b$ by $b$ .

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

Step 2.2.4

Rewrite using the commutative property of multiplication.

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

Step 2.2.5

Multiply $- 1$ by $- 1$ .

$a^{2} + a C - b^{2} + b C - c a + 1 c b - c C$

Step 2.2.6

Multiply $c$ by $1$ .

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

Step 2.3

Simplify the expression.

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Step 2.3.1

Reorder $a$ and $C$ .

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

Step 2.3.2

Reorder $b$ and $C$ .

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

Step 2.3.3

Move $c$ .

$a^{2} + C a - b^{2} + C b - 1 a c + c b - c C$

Step 2.3.4

Reorder $c$ and $b$ .

$a^{2} + C a - b^{2} + C b - 1 a c + b c - c C$

Step 2.3.5

Move $c$ .

$a^{2} + C a - b^{2} + C b - 1 a c + b c - 1 C c$

Step 2.3.6

Move $- b^{2}$ .

$a^{2} + C a + C b - 1 a c + b c - 1 C c - b^{2}$

Step 2.3.7

Move $a^{2}$ .

$C a + C b - 1 a c + b c - 1 C c + a^{2} - b^{2}$

Step 2.3.8

Move $b c$ .

$C a + C b - 1 a c - 1 C c + b c + a^{2} - b^{2}$

Step 2.3.9

Move $- 1 a c$ .

$C a + C b - 1 C c - 1 a c + b c + a^{2} - b^{2}$

Step 2.3.10

Move $C b$ .

$C a - 1 C c + C b - 1 a c + b c + a^{2} - b^{2}$

Step 2.3.11

Reorder $C a$ and $- 1 C c$ .

$- 1 C c + C a + C b - 1 a c + b c + a^{2} - b^{2}$