Question

Factorise :- x² - ax - (2a² - 3ab+b²)​

Answer 1

Answer:

$\boxed{\sf \: {x}^{2} - ax - ( {2a}^{2} - 3ab + {b}^{2}) = (x -2a + b) \: (x + a - b) \: }$

Step-by-step explanation:

Given algebraic expression is

$\sf \: {x}^{2} - ax - ( {2a}^{2} - 3ab + {b}^{2})$

$\sf \: = \: {x}^{2} - ax - [ {2a}^{2} - 2ab - ab+ {b}^{2}]$

$\sf \: = \: {x}^{2} - ax - [2a(a - b) - b(a - b)]$

$\sf \: = \: {x}^{2} - ax - (a - b)(2a - b)$

can be rewritten as

$\sf \: = \: {x}^{2} - (2a - a)x - (a - b)(2a - b)$

$\sf \: = \: {x}^{2} - (2a - a + b - b)x - (a - b)(2a - b)$

$\sf \: = \: {x}^{2} - [ (2a -b) -(a - b)]x - (a - b)(2a - b)$

$\sf \: = \: {x}^{2} -(2a -b)x + (a - b)x - (a - b)(2a - b)$

$\sf \: = \: x[ x -(2a -b)] + (a - b)[ x -(2a - b)]$

$\sf \: = \: [ x -(2a -b)] \: [ x + (a - b)]$

$\sf \: = \: (x -2a + b) \: (x + a - b)$

Hence,

$\boxed{\sf \: {x}^{2} - ax - ( {2a}^{2} - 3ab + {b}^{2}) = (x -2a + b) \: (x + a - b) \: }$

$\rule{190pt}{2pt}$

Additional Information

$\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{{More \: identities}}}} \\ \\ \bigstar \: \bf{ {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {x}^{2} - {y}^{2} = (x + y)(x - y)}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} - {(x - y)}^{2} = 4xy}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2})}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)}\:\\ \\ \bigstar \: \bf{ {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3xy(x - y) }\:\\ \\ \bigstar \: \bf{ {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} )}\: \end{array} }}\end{gathered}\end{gathered}\end{gathered}$