Expand the Trigonometric Expression (2a-3b+c)^2

${(2 a - 3 b + c)}^{2}$

Step 1

Rewrite ${(2 a - 3 b + c)}^{2}$ as $(2 a - 3 b + c) (2 a - 3 b + c)$ .

$(2 a - 3 b + c) (2 a - 3 b + c)$

Step 2

Expand $(2 a - 3 b + c) (2 a - 3 b + c)$ by multiplying each term in the first expression by each term in the second expression.

$2 a (2 a) + 2 a (- 3 b) + 2 a c - 3 b (2 a) - 3 b (- 3 b) - 3 b c + c (2 a) + c (- 3 b) + c \cdot c$

Step 3

Simplify each term.

Rewrite using the commutative property of multiplication.

$2 \cdot 2 a \cdot a + 2 a (- 3 b) + 2 a c - 3 b (2 a) - 3 b (- 3 b) - 3 b c + c (2 a) + c (- 3 b) + c \cdot c$

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

Move $a$ .

$2 \cdot 2 (a \cdot a) + 2 a (- 3 b) + 2 a c - 3 b (2 a) - 3 b (- 3 b) - 3 b c + c (2 a) + c (- 3 b) + c \cdot c$

Multiply $a$ by $a$ .

$2 \cdot 2 a^{2} + 2 a (- 3 b) + 2 a c - 3 b (2 a) - 3 b (- 3 b) - 3 b c + c (2 a) + c (- 3 b) + c \cdot c$

Multiply $2$ by $2$ .

$4 a^{2} + 2 a (- 3 b) + 2 a c - 3 b (2 a) - 3 b (- 3 b) - 3 b c + c (2 a) + c (- 3 b) + c \cdot c$

Rewrite using the commutative property of multiplication.

$4 a^{2} + 2 \cdot - 3 a b + 2 a c - 3 b (2 a) - 3 b (- 3 b) - 3 b c + c (2 a) + c (- 3 b) + c \cdot c$

Multiply $2$ by $- 3$ .

$4 a^{2} - 6 a b + 2 a c - 3 b (2 a) - 3 b (- 3 b) - 3 b c + c (2 a) + c (- 3 b) + c \cdot c$

Rewrite using the commutative property of multiplication.

$4 a^{2} - 6 a b + 2 a c - 3 \cdot 2 b a - 3 b (- 3 b) - 3 b c + c (2 a) + c (- 3 b) + c \cdot c$

Multiply $- 3$ by $2$ .

$4 a^{2} - 6 a b + 2 a c - 6 b a - 3 b (- 3 b) - 3 b c + c (2 a) + c (- 3 b) + c \cdot c$

Rewrite using the commutative property of multiplication.

$4 a^{2} - 6 a b + 2 a c - 6 b a - 3 \cdot - 3 b \cdot b - 3 b c + c (2 a) + c (- 3 b) + c \cdot c$

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

Move $b$ .

$4 a^{2} - 6 a b + 2 a c - 6 b a - 3 \cdot - 3 (b \cdot b) - 3 b c + c (2 a) + c (- 3 b) + c \cdot c$

Multiply $b$ by $b$ .

$4 a^{2} - 6 a b + 2 a c - 6 b a - 3 \cdot - 3 b^{2} - 3 b c + c (2 a) + c (- 3 b) + c \cdot c$

Multiply $- 3$ by $- 3$ .

$4 a^{2} - 6 a b + 2 a c - 6 b a + 9 b^{2} - 3 b c + c (2 a) + c (- 3 b) + c \cdot c$

Rewrite using the commutative property of multiplication.

$4 a^{2} - 6 a b + 2 a c - 6 b a + 9 b^{2} - 3 b c + 2 c a + c (- 3 b) + c \cdot c$

Rewrite using the commutative property of multiplication.

$4 a^{2} - 6 a b + 2 a c - 6 b a + 9 b^{2} - 3 b c + 2 c a - 3 c b + c \cdot c$

Multiply $c$ by $c$ .

$4 a^{2} - 6 a b + 2 a c - 6 b a + 9 b^{2} - 3 b c + 2 c a - 3 c b + c^{2}$

Step 4

Subtract $6 b a$ from $- 6 a b$ .

Move $b$ .

$4 a^{2} - 6 a b - 6 a b + 2 a c + 9 b^{2} - 3 b c + 2 c a - 3 c b + c^{2}$

Subtract $6 a b$ from $- 6 a b$ .

$4 a^{2} - 12 a b + 2 a c + 9 b^{2} - 3 b c + 2 c a - 3 c b + c^{2}$

Step 5

Add $2 a c$ and $2 c a$ .

Move $c$ .

$4 a^{2} - 12 a b + 9 b^{2} - 3 b c + 2 a c + 2 a c - 3 c b + c^{2}$

Add $2 a c$ and $2 a c$ .

$4 a^{2} - 12 a b + 9 b^{2} - 3 b c + 4 a c - 3 c b + c^{2}$

Step 6

Subtract $3 c b$ from $- 3 b c$ .

Move $c$ .

$4 a^{2} - 12 a b + 9 b^{2} - 3 b c - 3 b c + 4 a c + c^{2}$

Subtract $3 b c$ from $- 3 b c$ .

$4 a^{2} - 12 a b + 9 b^{2} - 6 b c + 4 a c + c^{2}$

Expand the Trigonometric Expression (2a-3b+c)^2

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