Question

Factorize (by spliting middle term) 2 (x+ y)² – 9 (x + y) -5​

Answer 1

Answer:

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Answer 2

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

Given expression is

$\rm :\longmapsto\:2 {(x + y)}^{2} - 9(x + y) - 5$

Let assume that,

$\red{\rm :\longmapsto\:x + y = z}$

So, above expression can be rewritten as

$\rm \: = \:\: {2z}^{2} - 9z - 5$

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.

So,

$\rm \: = \:\: {2z}^{2} - 10z + z - 5$

$\rm \: = \:\:2z(z - 5) + 1(z - 5)$

$\rm \: = \:\:(z - 5)(2z + 1)$

Now, Substitute the value of z, we get

$\rm \: = \:\:(x + y - 5)[2(x + y) + 1]$

$\rm \: = \:\:(x + y - 5)[2x + 2y+ 1]$

More Identities to know :

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

a² - b² = (a + b)(a - b)

(a + b)² = (a - b)² + 4ab

(a - b)² = (a + b)² - 4ab

(a + b)² + (a - b)² = 2(a² + b²)

(a + b)³ = a³ + b³ + 3ab(a + b)

(a - b)³ = a³ - b³ - 3ab(a - b)