Solution

\[ \begin{align*} \color{green} (a+b+c)^2 &= \color{green} a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \\ & a = x, \, b = -2y, \, c = 3z \\ \\ & = [(x)+(-2y)+(3z)]^2 \\ & = (x)^2 + (-2y)^2 + (3z)^2 + 2 \times x \times (-2y) + 2 \times (-2y) \times (3z) + 2 \times (3z) \times x \\ & = \color{red} x^2 + 4y^2 + 9z^2 - 4xy - 12yz + 6xz \end{align*} \]

Solution

\[ \begin{align*} \color{green} (a+b+c)^2 &= \color{green} a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \\ & a = -5x, \, b = 2y, \, c = z \\ \\ & = [(-5x)+(2y)+(z)]^2 \\ & = (-5x)^2 + (2y)^2 + (z)^2 + 2 \times (-5x) \times (2y) + 2 \times (2y) \times (z) + 2 \times (z) \times (-5x) \\ & = \color{red} 25x^2 + 4y^2 + z^2 - 20xy + 4yz - 10xz \end{align*} \]

Solution

\[ \begin{align*} \color{green} (a+b+c)^2 &= \color{green} a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \\ & a = 4x, \, b = y, \, c = -3z \\ \\ & = [(4x)+(y)+(-3z)]^2 \\ & = (4x)^2 + (y)^2 + (-3z)^2 + 2 \times 4x \times y + 2 \times y \times (-3z) + 2 \times (-3z) \times 4x \\ & = \color{red} 16x^2 + y^2 + 9z^2 + 8xy - 6yz - 24xz \end{align*} \]

Solution

\[ \begin{align*} \color{green} (a+b+c)^2 &= \color{green} a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \\ & a = 3a, \, b = -5b, \, c = -7c \\ \\ & = [(3a)+(-5b)+(-7c)]^2 \\ & = (3a)^2 + (-5b)^2 + (-7c)^2 + 2 \times 3a \times (-5b) + 2 \times (-5b) \times (-7c) + 2 \times (-7c) \times 3a \\ & = 9a^2 + 25b^2 + 49c^2 - 30ab + 70bc - 42ac \\ & = \color{red} 9a^2 + 25b^2 + 49c^2 - 30ab + 70bc - 42ac \end{align*} \]

Solution

\[ \begin{align*} \color{green} (a+b+c)^2 &= \color{green} a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \\ & a = -2a, \, b = -b, \, c = 3c \\ \\ & = [(-2a)+(-b)+(3c)]^2 \\ & = (-2a)^2 + (-b)^2 + (3c)^2 + 2 \times (-2a) \times (-b) + 2 \times (-b) \times (3c) + 2 \times (3c) \times (-2a) \\ & = \color{red} 4a^2 + b^2 + 9c^2 + 4ab - 6bc - 12ac \end{align*} \]

Solution

\[ \begin{align*} \color{green} (a+b+c)^2 &= \color{green} a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \\ & a = -a, \, b = 6b, \, c = -2c \\ \\ & = [(-a)+(6b)+(-2c)]^2 \\ & = (-a)^2 + (6b)^2 + (-2c)^2 + 2 \times (-a) \times 6b + 2 \times 6b \times (-2c) + 2 \times (-2c) \times (-a) \\ & = \color{red} a^2 + 36b^2 + 4c^2 - 12ab - 24bc + 4ac \end{align*} \]

Solution

\[ \begin{align*} \color{green} (a+b+c)^2 &= \color{green} a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \\ & a = 1, \, b = 2x, \, c = -3y \\ \\ & = [(1)+(2x)+(-3y)]^2 \\ & = (1)^2 + (2x)^2 + (-3y)^2 + 2 \times 1 \times 2x + 2 \times 2x \times (-3y) + 2 \times (-3y) \times 1 \\ & = \color{red} 1 + 4x^2 + 9y^2 + 4x - 12xy - 6y \end{align*} \]

Solution

\[ \begin{align*} \color{green} (a+b+c)^2 &= \color{green} a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \\ & a = 2x, \, b = -4y, \, c = -1 \\ \\ & = [(2x)+(-4y)+(-1)]^2 \\ & = (2x)^2 + (-4y)^2 + (-1)^2 + 2 \times 2x \times (-4y) + 2 \times (-4y) \times (-1) + 2 \times (-1) \times 2x \\ & = \color{red} 4x^2 + 16y^2 + 1 - 16xy + 8y - 4x \end{align*} \]

Solution

\[ \begin{align*} \color{green} (a+b+c)^2 &= \color{green} a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \\ & a = 6p, \, b = -5q, \, c = -4r \\ \\ & = [(6p)+(-5q)+(-4r)]^2 \\ & = (6p)^2 + (-5q)^2 + (-4r)^2 + 2 \times 6p \times (-5q) + 2 \times (-5q) \times (-4r) + 2 \times (-4r) \times 6p \\ & = \color{red} 36p^2 + 25q^2 + 16r^2 - 60pq + 40qr - 48pr \end{align*} \]

Solution

\[ \begin{align*} \color{green} (a+b+c)^2 &= \color{green} a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \\ & a = p, \, b = 5q, \, c = 2 \\ \\ & = [(p)+(5q)+(2)]^2 \\ & = (p)^2 + (5q)^2 + (2)^2 + 2 \times p \times 5q + 2 \times 5q \times 2 + 2 \times 2 \times p \\ & = \color{red} p^2 + 25q^2 + 4 + 10pq + 20q + 4p \end{align*} \]

Solution

\[ \begin{align*} \color{green} (a+b+c)^2 &= \color{green} a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \\ & a = 3x, \, b = -4y, \, c = -5 \\ \\ & = [(3x)+(-4y)+(-5)]^2 \\ & = (3x)^2 + (-4y)^2 + (-5)^2 + 2 \times 3x \times (-4y) + 2 \times (-4y) \times (-5) + 2 \times (-5) \times 3x \\ & = \color{red} 9x^2 + 16y^2 + 25 - 24xy + 40y - 30x \end{align*} \]

Solution

\[ \begin{align*} \color{green} (a+b+c)^2 &= \color{green} a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \\ & a = -2x, \, b = 6y, \, c = 4 \\ \\ & = [(-2x)+(6y)+(4)]^2 \\ & = (-2x)^2 + (6y)^2 + (4)^2 + 2 \times (-2x) \times 6y + 2 \times 6y \times 4 + 2 \times 4 \times (-2x) \\ & = \color{red} 4x^2 + 36y^2 + 16 - 24xy + 48y - 16x \end{align*} \]