Simplify (1/3A-1/4B+2/5c)^2

${(\frac{1}{3} A - \frac{1}{4} B + \frac{2}{5} c)}^{2}$

Step 1

Simplify terms.

Simplify each term.

Combine $\frac{1}{3}$ and $A$ .

${(\frac{A}{3} - \frac{1}{4} B + \frac{2}{5} c)}^{2}$

Combine $B$ and $\frac{1}{4}$ .

${(\frac{A}{3} - \frac{B}{4} + \frac{2}{5} c)}^{2}$

Combine $\frac{2}{5}$ and $c$ .

${(\frac{A}{3} - \frac{B}{4} + \frac{2 c}{5})}^{2}$

Rewrite ${(\frac{A}{3} - \frac{B}{4} + \frac{2 c}{5})}^{2}$ as $(\frac{A}{3} - \frac{B}{4} + \frac{2 c}{5}) (\frac{A}{3} - \frac{B}{4} + \frac{2 c}{5})$ .

$(\frac{A}{3} - \frac{B}{4} + \frac{2 c}{5}) (\frac{A}{3} - \frac{B}{4} + \frac{2 c}{5})$

Step 2

Expand $(\frac{A}{3} - \frac{B}{4} + \frac{2 c}{5}) (\frac{A}{3} - \frac{B}{4} + \frac{2 c}{5})$ by multiplying each term in the first expression by each term in the second expression.

$\frac{A}{3} \cdot \frac{A}{3} + \frac{A}{3} (- \frac{B}{4}) + \frac{A}{3} \cdot \frac{2 c}{5} - \frac{B}{4} \cdot \frac{A}{3} - \frac{B}{4} (- \frac{B}{4}) - \frac{B}{4} \cdot \frac{2 c}{5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Step 3

Simplify terms.

Simplify each term.

Multiply $\frac{A}{3} \cdot \frac{A}{3}$ .

Multiply $\frac{A}{3}$ by $\frac{A}{3}$ .

$\frac{A \cdot A}{3 \cdot 3} + \frac{A}{3} (- \frac{B}{4}) + \frac{A}{3} \cdot \frac{2 c}{5} - \frac{B}{4} \cdot \frac{A}{3} - \frac{B}{4} (- \frac{B}{4}) - \frac{B}{4} \cdot \frac{2 c}{5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Raise $A$ to the power of $1$ .

$\frac{A^{1} A}{3 \cdot 3} + \frac{A}{3} (- \frac{B}{4}) + \frac{A}{3} \cdot \frac{2 c}{5} - \frac{B}{4} \cdot \frac{A}{3} - \frac{B}{4} (- \frac{B}{4}) - \frac{B}{4} \cdot \frac{2 c}{5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Raise $A$ to the power of $1$ .

$\frac{A^{1} A^{1}}{3 \cdot 3} + \frac{A}{3} (- \frac{B}{4}) + \frac{A}{3} \cdot \frac{2 c}{5} - \frac{B}{4} \cdot \frac{A}{3} - \frac{B}{4} (- \frac{B}{4}) - \frac{B}{4} \cdot \frac{2 c}{5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{A^{1 + 1}}{3 \cdot 3} + \frac{A}{3} (- \frac{B}{4}) + \frac{A}{3} \cdot \frac{2 c}{5} - \frac{B}{4} \cdot \frac{A}{3} - \frac{B}{4} (- \frac{B}{4}) - \frac{B}{4} \cdot \frac{2 c}{5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Add $1$ and $1$ .

$\frac{A^{2}}{3 \cdot 3} + \frac{A}{3} (- \frac{B}{4}) + \frac{A}{3} \cdot \frac{2 c}{5} - \frac{B}{4} \cdot \frac{A}{3} - \frac{B}{4} (- \frac{B}{4}) - \frac{B}{4} \cdot \frac{2 c}{5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Multiply $3$ by $3$ .

$\frac{A^{2}}{9} + \frac{A}{3} (- \frac{B}{4}) + \frac{A}{3} \cdot \frac{2 c}{5} - \frac{B}{4} \cdot \frac{A}{3} - \frac{B}{4} (- \frac{B}{4}) - \frac{B}{4} \cdot \frac{2 c}{5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Rewrite using the commutative property of multiplication.

$\frac{A^{2}}{9} - \frac{A}{3} \cdot \frac{B}{4} + \frac{A}{3} \cdot \frac{2 c}{5} - \frac{B}{4} \cdot \frac{A}{3} - \frac{B}{4} (- \frac{B}{4}) - \frac{B}{4} \cdot \frac{2 c}{5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Multiply $- \frac{A}{3} \cdot \frac{B}{4}$ .

Multiply $\frac{B}{4}$ by $\frac{A}{3}$ .

$\frac{A^{2}}{9} - \frac{B A}{4 \cdot 3} + \frac{A}{3} \cdot \frac{2 c}{5} - \frac{B}{4} \cdot \frac{A}{3} - \frac{B}{4} (- \frac{B}{4}) - \frac{B}{4} \cdot \frac{2 c}{5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Multiply $4$ by $3$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{A}{3} \cdot \frac{2 c}{5} - \frac{B}{4} \cdot \frac{A}{3} - \frac{B}{4} (- \frac{B}{4}) - \frac{B}{4} \cdot \frac{2 c}{5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Multiply $\frac{A}{3} \cdot \frac{2 c}{5}$ .

Multiply $\frac{A}{3}$ by $\frac{2 c}{5}$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{A (2 c)}{3 \cdot 5} - \frac{B}{4} \cdot \frac{A}{3} - \frac{B}{4} (- \frac{B}{4}) - \frac{B}{4} \cdot \frac{2 c}{5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Multiply $3$ by $5$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{A (2 c)}{15} - \frac{B}{4} \cdot \frac{A}{3} - \frac{B}{4} (- \frac{B}{4}) - \frac{B}{4} \cdot \frac{2 c}{5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Move $2$ to the left of $A$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 \cdot A c}{15} - \frac{B}{4} \cdot \frac{A}{3} - \frac{B}{4} (- \frac{B}{4}) - \frac{B}{4} \cdot \frac{2 c}{5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Multiply $- \frac{B}{4} \cdot \frac{A}{3}$ .

Multiply $\frac{A}{3}$ by $\frac{B}{4}$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{3 \cdot 4} - \frac{B}{4} (- \frac{B}{4}) - \frac{B}{4} \cdot \frac{2 c}{5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Multiply $3$ by $4$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} - \frac{B}{4} (- \frac{B}{4}) - \frac{B}{4} \cdot \frac{2 c}{5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Multiply $- \frac{B}{4} (- \frac{B}{4})$ .

Multiply $- 1$ by $- 1$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + 1 \frac{B}{4} \frac{B}{4} - \frac{B}{4} \cdot \frac{2 c}{5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Multiply $\frac{B}{4}$ by $1$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B}{4} \cdot \frac{B}{4} - \frac{B}{4} \cdot \frac{2 c}{5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Multiply $\frac{B}{4}$ by $\frac{B}{4}$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B \cdot B}{4 \cdot 4} - \frac{B}{4} \cdot \frac{2 c}{5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Raise $B$ to the power of $1$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{1} B}{4 \cdot 4} - \frac{B}{4} \cdot \frac{2 c}{5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Raise $B$ to the power of $1$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{1} B^{1}}{4 \cdot 4} - \frac{B}{4} \cdot \frac{2 c}{5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{1 + 1}}{4 \cdot 4} - \frac{B}{4} \cdot \frac{2 c}{5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Add $1$ and $1$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{2}}{4 \cdot 4} - \frac{B}{4} \cdot \frac{2 c}{5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Multiply $4$ by $4$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{2}}{16} - \frac{B}{4} \cdot \frac{2 c}{5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{B}{4}$ into the numerator.

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{2}}{16} + \frac{- B}{4} \cdot \frac{2 c}{5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Factor $2$ out of $4$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{2}}{16} + \frac{- B}{2 (2)} \cdot \frac{2 c}{5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Factor $2$ out of $2 c$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{2}}{16} + \frac{- B}{2 (2)} \cdot \frac{2 (c)}{5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Cancel the common factor.

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{2}}{16} + \frac{- B}{2 \cdot 2} \cdot \frac{2 c}{5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Rewrite the expression.

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{2}}{16} + \frac{- B}{2} \cdot \frac{c}{5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Multiply $\frac{- B}{2}$ by $\frac{c}{5}$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{2}}{16} + \frac{- B c}{2 \cdot 5} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Multiply $2$ by $5$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{2}}{16} + \frac{- B c}{10} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Move the negative in front of the fraction.

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{2}}{16} - \frac{B c}{10} + \frac{2 c}{5} \cdot \frac{A}{3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Multiply $\frac{2 c}{5} \cdot \frac{A}{3}$ .

Multiply $\frac{2 c}{5}$ by $\frac{A}{3}$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{2}}{16} - \frac{B c}{10} + \frac{2 c A}{5 \cdot 3} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Multiply $5$ by $3$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{2}}{16} - \frac{B c}{10} + \frac{2 c A}{15} + \frac{2 c}{5} (- \frac{B}{4}) + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Rewrite using the commutative property of multiplication.

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{2}}{16} - \frac{B c}{10} + \frac{2 c A}{15} - \frac{2 c}{5} \cdot \frac{B}{4} + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{2 c}{5}$ into the numerator.

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{2}}{16} - \frac{B c}{10} + \frac{2 c A}{15} + \frac{- 2 c}{5} \cdot \frac{B}{4} + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Factor $2$ out of $- 2 c$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{2}}{16} - \frac{B c}{10} + \frac{2 c A}{15} + \frac{2 (- c)}{5} \cdot \frac{B}{4} + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Factor $2$ out of $4$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{2}}{16} - \frac{B c}{10} + \frac{2 c A}{15} + \frac{2 (- c)}{5} \cdot \frac{B}{2 (2)} + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Cancel the common factor.

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{2}}{16} - \frac{B c}{10} + \frac{2 c A}{15} + \frac{2 (- c)}{5} \cdot \frac{B}{2 \cdot 2} + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Rewrite the expression.

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{2}}{16} - \frac{B c}{10} + \frac{2 c A}{15} + \frac{- c}{5} \cdot \frac{B}{2} + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Multiply $\frac{- c}{5}$ by $\frac{B}{2}$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{2}}{16} - \frac{B c}{10} + \frac{2 c A}{15} + \frac{- c B}{5 \cdot 2} + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Multiply $5$ by $2$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{2}}{16} - \frac{B c}{10} + \frac{2 c A}{15} + \frac{- c B}{10} + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Move the negative in front of the fraction.

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{2}}{16} - \frac{B c}{10} + \frac{2 c A}{15} - \frac{c B}{10} + \frac{2 c}{5} \cdot \frac{2 c}{5}$

Multiply $\frac{2 c}{5} \cdot \frac{2 c}{5}$ .

Multiply $\frac{2 c}{5}$ by $\frac{2 c}{5}$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{2}}{16} - \frac{B c}{10} + \frac{2 c A}{15} - \frac{c B}{10} + \frac{2 c (2 c)}{5 \cdot 5}$

Multiply $2$ by $2$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{2}}{16} - \frac{B c}{10} + \frac{2 c A}{15} - \frac{c B}{10} + \frac{4 c \cdot c}{5 \cdot 5}$

Raise $c$ to the power of $1$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{2}}{16} - \frac{B c}{10} + \frac{2 c A}{15} - \frac{c B}{10} + \frac{4 (c^{1} c)}{5 \cdot 5}$

Raise $c$ to the power of $1$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{2}}{16} - \frac{B c}{10} + \frac{2 c A}{15} - \frac{c B}{10} + \frac{4 (c^{1} c^{1})}{5 \cdot 5}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{2}}{16} - \frac{B c}{10} + \frac{2 c A}{15} - \frac{c B}{10} + \frac{4 c^{1 + 1}}{5 \cdot 5}$

Add $1$ and $1$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{2}}{16} - \frac{B c}{10} + \frac{2 c A}{15} - \frac{c B}{10} + \frac{4 c^{2}}{5 \cdot 5}$

Multiply $5$ by $5$ .

$\frac{A^{2}}{9} - \frac{B A}{12} + \frac{2 A c}{15} - \frac{A B}{12} + \frac{B^{2}}{16} - \frac{B c}{10} + \frac{2 c A}{15} - \frac{c B}{10} + \frac{4 c^{2}}{25}$

Combine the numerators over the common denominator.

$\frac{A^{2}}{9} + \frac{- B A - A B}{12} + \frac{2 A c + 2 c A}{15} + \frac{B^{2}}{16} + \frac{- B c - c B}{10} + \frac{4 c^{2}}{25}$

Step 4

Subtract $A B$ from $- B A$ .

Move $B$ .

$\frac{A^{2}}{9} + \frac{- A B - A B}{12} + \frac{2 A c + 2 c A}{15} + \frac{B^{2}}{16} + \frac{- B c - c B}{10} + \frac{4 c^{2}}{25}$

Subtract $A B$ from $- A B$ .

$\frac{A^{2}}{9} + \frac{- 2 A B}{12} + \frac{2 A c + 2 c A}{15} + \frac{B^{2}}{16} + \frac{- B c - c B}{10} + \frac{4 c^{2}}{25}$

Step 5

Add $2 A c$ and $2 c A$ .

Move $A$ .

$\frac{A^{2}}{9} + \frac{- 2 A B}{12} + \frac{2 c A + 2 c A}{15} + \frac{B^{2}}{16} + \frac{- B c - c B}{10} + \frac{4 c^{2}}{25}$

Add $2 c A$ and $2 c A$ .

$\frac{A^{2}}{9} + \frac{- 2 A B}{12} + \frac{4 c A}{15} + \frac{B^{2}}{16} + \frac{- B c - c B}{10} + \frac{4 c^{2}}{25}$

Step 6

Subtract $c B$ from $- B c$ .

Move $B$ .

$\frac{A^{2}}{9} + \frac{- 2 A B}{12} + \frac{4 c A}{15} + \frac{B^{2}}{16} + \frac{- c B - c B}{10} + \frac{4 c^{2}}{25}$

Subtract $c B$ from $- c B$ .

$\frac{A^{2}}{9} + \frac{- 2 A B}{12} + \frac{4 c A}{15} + \frac{B^{2}}{16} + \frac{- 2 c B}{10} + \frac{4 c^{2}}{25}$

Step 7

Simplify each term.

Cancel the common factor of $- 2$ and $12$ .

Factor $2$ out of $- 2 A B$ .

$\frac{A^{2}}{9} + \frac{2 (- A B)}{12} + \frac{4 c A}{15} + \frac{B^{2}}{16} + \frac{- 2 c B}{10} + \frac{4 c^{2}}{25}$

Cancel the common factors.

Factor $2$ out of $12$ .

$\frac{A^{2}}{9} + \frac{2 (- A B)}{2 (6)} + \frac{4 c A}{15} + \frac{B^{2}}{16} + \frac{- 2 c B}{10} + \frac{4 c^{2}}{25}$

Cancel the common factor.

$\frac{A^{2}}{9} + \frac{2 (- A B)}{2 \cdot 6} + \frac{4 c A}{15} + \frac{B^{2}}{16} + \frac{- 2 c B}{10} + \frac{4 c^{2}}{25}$

Rewrite the expression.

$\frac{A^{2}}{9} + \frac{- A B}{6} + \frac{4 c A}{15} + \frac{B^{2}}{16} + \frac{- 2 c B}{10} + \frac{4 c^{2}}{25}$

Move the negative in front of the fraction.

$\frac{A^{2}}{9} - \frac{A B}{6} + \frac{4 c A}{15} + \frac{B^{2}}{16} + \frac{- 2 c B}{10} + \frac{4 c^{2}}{25}$

Cancel the common factor of $- 2$ and $10$ .

Factor $2$ out of $- 2 c B$ .

$\frac{A^{2}}{9} - \frac{A B}{6} + \frac{4 c A}{15} + \frac{B^{2}}{16} + \frac{2 (- c B)}{10} + \frac{4 c^{2}}{25}$

Cancel the common factors.

Factor $2$ out of $10$ .

$\frac{A^{2}}{9} - \frac{A B}{6} + \frac{4 c A}{15} + \frac{B^{2}}{16} + \frac{2 (- c B)}{2 (5)} + \frac{4 c^{2}}{25}$

Cancel the common factor.

$\frac{A^{2}}{9} - \frac{A B}{6} + \frac{4 c A}{15} + \frac{B^{2}}{16} + \frac{2 (- c B)}{2 \cdot 5} + \frac{4 c^{2}}{25}$

Rewrite the expression.

$\frac{A^{2}}{9} - \frac{A B}{6} + \frac{4 c A}{15} + \frac{B^{2}}{16} + \frac{- c B}{5} + \frac{4 c^{2}}{25}$

Move the negative in front of the fraction.

$\frac{A^{2}}{9} - \frac{A B}{6} + \frac{4 c A}{15} + \frac{B^{2}}{16} - \frac{c B}{5} + \frac{4 c^{2}}{25}$

Simplify (1/3A-1/4B+2/5c)^2

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