25(2x-y)²-81y² factorise . please give an appropriate answer

1.	25(2x-y)²-81y² factorise . please give an appropriate answer
Answer» ANSWER :- $\implies\sf 25(2x - y)^2 - 81y^2$ $\implies\sf [5(2x - y)]^2 - (9y)^2$ $\implies\sf [10x - 5y]^2 - ( 9y)^2$ USING the identity :- $\sf a^2 - b^2 = (a+b)(a-b)$ $\implies\sf (10x - 5y + 9y)(10x - 5y - 9y)$ $\implies\sf (10x + 4y)(10x - 14y)$ $\boxed{\sf 25(2x - y)^2 - 81y^2 = (10x + 4y)(10x - 14y)}$ VERIFICATION :- $\sf <klux>LHS</klux> = 25(2x - y)^2 - 81y^2$ $\implies\sf LHS = 25(4x^2 + y^2 - 4xy) - 81y^2$ $\implies\sf LHS = 100x^2 + 25y^2 - 100xy - 81y^2$ $\implies\sf LHS = 100x^2 - 56y^2 - 100xy$ $\sf RHS = ( 10x + 4y)(10x - 14y)$ $\implies\sf RHS = 10x(10x - 14y) + 4y(10x - 14y)$ $\implies\sf RHS = 100x^2 - 140xy + 40xy - 56y^2$ $\implies\sf RHS = 100x^2 - 100xy - 56y^2$ LHS = RHS Hence verified. ADDITIONAL information :- $\boxed{\bigstar\:\:\textbf{\textsf{Algebraic\:Identity}}\:\bigstar}\\\\1)\sf\:(A+B)^{2} = A^{2} + 2AB + B^{2}\\\\2)\sf\: (A-B)^{2} = A^{2} - 2AB + B^{2}\\\\3)\sf\: A^{2} - B^{2} = (A+B)(A-B)\\\\4)\sf\: (A+B)^{2} = (A-B)^{2} + 4AB\\\\5)\sf\: (A-B)^{2} = (A+B)^{2} - 4AB\\\\6)\sf\: (A+B)^{3} = A^{3} + 3AB(A+B) + B^{3}\\\\7)\sf\:(A-B)^{3} = A^{3} - 3AB(A-B) + B^{3}\\\\8)\sf\: A^{3} + B^{3} = (A+B)(A^{2} - AB + B^{2}$

Answer»

ANSWER :-

$\implies\sf 25(2x - y)^2 - 81y^2$

$\implies\sf [5(2x - y)]^2 - (9y)^2$

$\implies\sf [10x - 5y]^2 - ( 9y)^2$

USING the identity :-

$\sf a^2 - b^2 = (a+b)(a-b)$

$\implies\sf (10x - 5y + 9y)(10x - 5y - 9y)$

$\implies\sf (10x + 4y)(10x - 14y)$

$\boxed{\sf 25(2x - y)^2 - 81y^2 = (10x + 4y)(10x - 14y)}$

VERIFICATION :-

$\sf <klux>LHS</klux> = 25(2x - y)^2 - 81y^2$

$\implies\sf LHS = 25(4x^2 + y^2 - 4xy) - 81y^2$

$\implies\sf LHS = 100x^2 + 25y^2 - 100xy - 81y^2$

$\implies\sf LHS = 100x^2 - 56y^2 - 100xy$

$\sf RHS = ( 10x + 4y)(10x - 14y)$

$\implies\sf RHS = 10x(10x - 14y) + 4y(10x - 14y)$

$\implies\sf RHS = 100x^2 - 140xy + 40xy - 56y^2$

$\implies\sf RHS = 100x^2 - 100xy - 56y^2$

LHS = RHS

Hence verified.

ADDITIONAL information :-

$\boxed{\bigstar\:\:\textbf{\textsf{Algebraic\:Identity}}\:\bigstar}\\\\1)\sf\:(A+B)^{2} = A^{2} + 2AB + B^{2}\\\\2)\sf\: (A-B)^{2} = A^{2} - 2AB + B^{2}\\\\3)\sf\: A^{2} - B^{2} = (A+B)(A-B)\\\\4)\sf\: (A+B)^{2} = (A-B)^{2} + 4AB\\\\5)\sf\: (A-B)^{2} = (A+B)^{2} - 4AB\\\\6)\sf\: (A+B)^{3} = A^{3} + 3AB(A+B) + B^{3}\\\\7)\sf\:(A-B)^{3} = A^{3} - 3AB(A-B) + B^{3}\\\\8)\sf\: A^{3} + B^{3} = (A+B)(A^{2} - AB + B^{2}$