Basic Math Examples

$(1 - a + b) (b - a + 1)$

Step 1

Expand $(1 - a + b) (b - a + 1)$ by multiplying each term in the first expression by each term in the second expression.

$1 b + 1 (- a) + 1 \cdot 1 - a b - a (- a) - a \cdot 1 + b \cdot b + b (- a) + b \cdot 1$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $b$ by $1$ .

$b + 1 (- a) + 1 \cdot 1 - a b - a (- a) - a \cdot 1 + b \cdot b + b (- a) + b \cdot 1$

Step 2.1.2

Multiply $- a$ by $1$ .

$b - a + 1 \cdot 1 - a b - a (- a) - a \cdot 1 + b \cdot b + b (- a) + b \cdot 1$

Step 2.1.3

Multiply $1$ by $1$ .

$b - a + 1 - a b - a (- a) - a \cdot 1 + b \cdot b + b (- a) + b \cdot 1$

Step 2.1.4

Rewrite using the commutative property of multiplication.

$b - a + 1 - a b - 1 \cdot - 1 a \cdot a - a \cdot 1 + b \cdot b + b (- a) + b \cdot 1$

Step 2.1.5

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

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Step 2.1.5.1

Move $a$ .

$b - a + 1 - a b - 1 \cdot - 1 (a \cdot a) - a \cdot 1 + b \cdot b + b (- a) + b \cdot 1$

Step 2.1.5.2

Multiply $a$ by $a$ .

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

Step 2.1.6

Multiply $- 1$ by $- 1$ .

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

Step 2.1.7

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

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

Step 2.1.8

Multiply $- 1$ by $1$ .

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

Step 2.1.9

Multiply $b$ by $b$ .

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

Step 2.1.10

Rewrite using the commutative property of multiplication.

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

Step 2.1.11

Multiply $b$ by $1$ .

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

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Add $b$ and $b$ .

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

Step 2.2.2

Subtract $a$ from $- a$ .

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

Step 3

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

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Step 3.1

Move $b$ .

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

Step 3.2

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

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

Step 4

Move $1$ .

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

Step 5

Move $- 2 a$ .

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

Step 6

Move $b^{2}$ .

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