Solution

\[ \begin{align*} & \color{green} a^2 + b^2 - 2ab = (a-b)^2 \\[4pt] & =x^2 + z^2 -2xz \\[4pt] &= (x)^2 + (z)^2 - 2 (x) (z) \\[4pt] & = (x-z)^2 \\[4pt] & = \color{red} (x-z)(x-z) \end{align*} \]

Solution

\[ \begin{align*} & \color{green} a^2 + b^2 - 2ab = (a-b)^2 \\[4pt] & = 4x^2 + 9y^2 - 12xy \\[4pt] & = (2x)^2 + (3y)^2 - 2 (2x) (3y) \\[4pt] & = (2x-3y)^2 \\[4pt] & = \color{red} (2x-3y)(2x-3y) \end{align*} \]

Solution

\[ \begin{align*} & \color{green} a^2 + b^2 + 2ab = (a+b)^2 \\[4pt] &= 64a^2 + 49b^2 + 112ab \\[4pt] &= (8a)^2 + (7b)^2 + 2 (8a) (7b) \\[4pt] & = (8a + 7b)^2 \\[4pt] & = \color{red} (8a + 7b)(8a + 7b) \end{align*} \]

Solution

\[ \begin{align*} & \color{green} a^2 + b^2 - 2ab = ( a-b)^2 \\[4pt] & = 121p^2 + 16q^2 - 88pq \\[4pt] &= (11p)^2 + (4q)^2 - 2 (11p) (4q) \\[4pt] & = (11p - 4q)^2 \\[4pt] & = \color{red} (11p - 4q)(11p - 4q) \end{align*} \]

Solution

\[ \begin{align*} & \color{green} a^2 - 2ab + b^2 = (a-b)^2 \\[4pt] & = 9x^2y - 24xy^2 + 16y^3 \\[4pt] &= y(9x^2 - 24xy + 16y^2) \\[4pt] &= y[(3x)^2 - 2 (3x) (4y)+ (4y)^2] \\[4pt] &= y(3x- 4y)^2 \\[4pt] & = \color{red} y(3x - 4y)(3x - 4y) \end{align*} \]

Solution

\[ \begin{align*} & \color{green} a^2 + 2ab + b^2 = \color{green} (a+b)^2 \\[4pt] & = 2a^3 + 4a^2b + 2ab^2 \\[4pt] &= 2a(a^2 + 2ab + b^2) \\[4pt] &= 2a[(a)^2 +2 (a) (b) + (b)^2] \\[4pt] &= 2a(a + b)^2 \\[4pt] & = \color{red} 2a(a+b)(a+b) \end{align*} \]

Solution

\[ \begin{align*} & = 50a^2 + 98b^2 + 140ab \\[4pt] &= 2[25a^2 + 49b^2 + 70ab] \\[4pt] &= 2[(5a)^2 + (7b)^2 + 2 (5a) (7b) ] \\[4pt] & \color{green} a^2 + b^2 + 2ab = \color{green} (a+b)^2 \\[4pt]& = 2[(5a + 7b)^2] \\[4pt] & = \color{red} 2(5a + 7b)(5a + 7b) \end{align*} \]

Solution

\[ \begin{align*} & \color{green} a^2 - b^2 = (a-b)(a+b) \\[4pt] & = 64x^2 - 81y^2 + 2ab \\[4pt] &= (8x)^2 - (9y)^2 \\[4pt] & = \color{red} (8x - 9y)(8x + 9y) \end{align*} \]

Solution

\[ \begin{align*} & \color{green} a^2 - b^2 = (a-b)(a+b) \\[4pt] & = 25p^2 - 9q^2 \\[4pt] & = (5p)^2 - (3q)^2 \\[4pt] & = \color{red} (5p - 3q)(5p + 3q) \end{align*} \]

Solution

\[ \begin{align*} & \color{green} a^2 - b^2 = (a-b)(a+b) \\[4pt] &= 16a^2b - 64b^3 \\[4pt] &= 16b(a^2 - 4b^2) \\[4pt] &= 16b[(a)^2 - (2b)^2] \\[4pt] & = \color{red} 16b(a - 2b)(a + 2b) \end{align*} \]

Solution

\[ \begin{align*} & \color{green} a^2 - b^2 = (a-b)(a+b) \\[4pt] & = 25x^3y^3 - 49xy \\[4pt] &= xy(25x^2y^2 - 49) \\[4pt] &= xy[(5xy)^2 - (7)^2] \\[4pt] & = \color{red} xy(5xy - 7)(5xy + 7) \end{align*} \]

Solution

\[ \begin{align*} & \color{green} a^2 - b^2 = (a-b)(a+b) \\[4pt] & = p^4 - 256 \\[4pt] &= (p^2)^2 - (16)^2 \\[4pt] &= (p^2 + 16)(p^2 - 16) \\[4pt] &= (p^2 + 16)[(p)^2 - (4)^2] \\[4pt] & = \color{red} (p^2 + 16) (p - 4)(p + 4) \end{align*} \]

Solution

\[ \begin{align*} & \color{green} a^2 - b^2 = (a-b)(a+b) \\[4pt] & = a^2 - (b-c)^2 \\[4pt] &= a^2 - [(b-c)]^2 \\[4pt] & = [a + (b-c)][a - (b-c)] \\[4pt] & = \color{red} (a + b - c)(a - b + c) \end{align*} \]

Solution

\[ \begin{align*} & \color{green} a^2 - b^2 = (a-b)(a+b) \\[4pt] & = 25m^2 - (4n + 3l)^2 \\[4pt] & = (5m)^2 - (4n + 3l)^2 \\[4pt] & = [5m - (4n + 3l)][5m + (4n + 3l)] \\[4pt] & = \color{red} (5m - 4n - 3l)(5m + 4n + 3l) \end{align*} \]

Solution

\[ \begin{align*} & \color{green} a^2 - b^2 = (a-b)(a+b) \\[4pt] & = (2a + 3b)^2 - 4c^2 \\[4pt] & = [(2a + 3b)]^2 - (2c)^2 \\[4pt] & = \color{red} (2a + 3b - 2c)(2a + 3b + 2c) \end{align*} \]

Solution

\[ \begin{align*} & \color{green} a^2 - 2ab + b^2 = (a-b)^2 \\[4pt] & = (64m^2 - 144mn + 81n^2) - 25p^2 \\[4pt] & = [(8m)^2 - 2 (8m) (9n)+ (9n)^2] - (5p)^2 \\[4pt] &= (8m - 9n)^2 - (5p)^2 \\[4pt] &\color{green} a^2 - b^2 = (a-b)(a+b) \\[4pt] & = \color{red} (8m - 9n - 5p)(8m - 9n + 5p) \end{align*} \]

Solution

\[ \begin{align*} & \implies \color{green} a^2 + b^2 + c^2 + 2ab + 2bc + 2ca = (a+b+c)^2 \\[4pt] &= 16x^2 + 9y^2 + 4z^2 + 24xy + 12yz + 16zx \\[4pt] &= (4x)^2 + (3y)^2 + (2z)^2 + 2 (4x) (3y) + 2 (3y) (2z) + 2 (2z) (4x) \\[4pt] & = (4x + 3y + 2z)^2 \\[4pt] & = \color{red} (4x + 3y + 2z)(4x + 3y + 2z) \end{align*} \]

Solution

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

Solution

\[ \begin{align*} &\implies \color{green} a^2 + b^2 + c^2 + 2ab + 2bc + 2ca = (a+b+c)^2 \\[4pt] &= (2a)^2 + (-b)^2 + (5c)^2 + 2(2a)(-b) - 2 (-b) (5c) + 2 (5c) (2a) \\[4pt] & = (2a - b + 5c)^2 \\[4pt] & = \color{red} (2a - b + 5c)(2a - b + 5c) \end{align*} \]

Solution

\[ \begin{align*} &\implies \color{green} a^2 + b^2 + c^2 + 2ab + 2bc + 2ca = (a+b+c)^2 \\[6pt] & = a^2 + \frac{b^2}{4} + \frac{c^2}{9} + ab + \frac{bc}{3} + \frac{2ca}{3} \\[6pt] &= (a)^2 + \left( \frac{b}{2} \right)^2 + \left( \frac{c}{3} \right)^2 + 2 (a) \left(\frac{b}{2}\right) + 2 \left(\frac{b}{2}\right) \left(\frac{c}{3}\right) + 2 \left(\frac{c}{3}\right) (a) \\[6pt] & = \left( a + \frac{b}{2} + \frac{c}{3} \right)^2 \\[6pt] & = \color{red} \left( a + \frac{b}{2} + \frac{c}{3} \right) \left( a + \frac{b}{2} + \frac{c}{3} \right) \end{align*} \]