Question

help me to solve the expression a² + ab- 2b²​

Answer 1

Answer:

a^2+ab-2b^2

=a^2+3ab-2ab-3b^2+b^2

=a^2-2ab+b^2+3ab-3b^2

=(a-b) ^2+3b(a-b)

=(a-b) (a-b+3b)

=(a-b) (a+2b)

Answer 2

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm :\longmapsto\: {a}^{2} + ab - {2b}^{2}$

Method used :-

Splitting of middle terms :-

In order to factorize  ax² + bx + c we have to find numbers p and q such that p + q = b and pq = ac.

After finding p and q, we split the middle term in the quadratic as px + qx and get required factors by grouping the terms.

❥ Solution :-

$\rm :\longmapsto\: {a}^{2} + ab - {2b}^{2}$

So,

Here, we have to find p and q such that

$\rm :\longmapsto\:pq = - 2 {a}^{2} {b}^{2} \: and \: p + q = ab$

Now,

pq can be rewritten as

$\rm :\longmapsto\:pq = - 2 {a}^{2} {b}^{2} = (2ab) \times ( - ab)$

so that

$\rm :\longmapsto\:p + q = 2ab - ab = ab$

So,

$\rm :\longmapsto\:p = 2ab \: and \: q = - \: ab$

Thus,

Given expression,

$\rm :\longmapsto\: {a}^{2} + ab - {2b}^{2}$

can be rewritten as

$\rm \: = \: \:\: {a}^{2} + 2ab - ab - {2b}^{2}$

$\rm \: = \: \:a(a + 2b) - b(a + 2b)$

$\rm \: = \: \:(a + 2b)(a - b)$

Hence,

$\: \: \: \: \: \: \underbrace{\boxed{ \sf{ \:\rm \: {a}^{2} + ab - 2 {b}^{2} = \: \:(a + 2b)(a - b)}}}$

Additional Information :-

Let's solve more problems of same type!!

Question :- Factorize the following

Q - 1

$\rm :\longmapsto\: {x}^{2} - xy - {2y}^{2}$

$\rm \: = \: \: {x}^{2} - 2xy + xy - {2y}^{2}$

$\rm \: = \: \:x(x - 2y) + y(x - 2y)$

$\rm \: = \: \:(x - 2y)(x + y)$

Q - 2

$\rm :\longmapsto\: {2x}^{2} + xy - 3 {y}^{2}$

$\rm \: = \: \: {2x}^{2} + 3xy - 2xy - {3y}^{2}$

$\rm \: = \: \:x(2x + 3y) - y(2x + 3y)$

$\rm \: = \: \:(2x + 3y)(x - y)$

Q - 3

$\rm :\longmapsto\: {3x}^{2} - xy - {2y}^{2}$

$\rm \: = \: \: {3x}^{2} - 3xy + 2xy - {2y}^{2}$

$\rm \: = \: \:3x(x - y) + 2y(x - y)$

$\rm \: = \: \:(3x + 2y)(x - y)$