Simplify (a-b+c)^2

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

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

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

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

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

Simplify each term.

Multiply $a$ by $a$ .

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

Rewrite using the commutative property of multiplication.

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

Rewrite using the commutative property of multiplication.

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

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

Move $b$ .

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

Multiply $b$ by $b$ .

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

Multiply $- 1$ by $- 1$ .

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

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

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

Rewrite using the commutative property of multiplication.

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

Multiply $c$ by $c$ .

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

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

Move $b$ .

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

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

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

Add $a c$ and $c a$ .

Reorder $c$ and $a$ .

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

Add $a c$ and $a c$ .

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

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

Move $c$ .

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

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

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

Move $- 2 b c$ .

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

Move $b^{2}$ .

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