Basic Math Examples

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

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Subtract $a b$ from $a b$ .

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

Step 2.1.3

Add $a c + a (- a) + b \cdot b + b c$ and $0$ .

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

Step 2.1.4

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

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

Step 2.1.5

Subtract $a c$ from $a c$ .

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

Step 2.1.6

Add $a (- a) + b \cdot b + b c + c b + c \cdot c$ and $0$ .

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

Step 2.2

Simplify each term.

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Step 2.2.1

Rewrite using the commutative property of multiplication.

$- a \cdot a + b \cdot b + b c + c b + c \cdot c$

Step 2.2.2

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

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Step 2.2.2.1

Move $a$ .

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

Step 2.2.2.2

Multiply $a$ by $a$ .

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

Step 2.2.3

Multiply $b$ by $b$ .

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

Step 2.2.4

Multiply $c$ by $c$ .

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

Step 3

Add $b c$ and $c b$ .

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Step 3.1

Reorder $c$ and $b$ .

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

Step 3.2

Add $b c$ and $b c$ .

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