Question

Factorise using middle term splitting

Its a little hard

5(x-y)^2-38(x-y)+21

Answer 1

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

Given expression is

$\rm :\longmapsto\: {5(x - y)}^{2} - 38(x - y) + 21$

Let assume that

$\rm :\longmapsto\:\boxed{ \tt{ \: x - y = z \: }} - - - - (1)$

So, above expression can be rewritten as

$\rm \:  =  \: {5z}^{2} - 38z + 21$

Using splitting of middle terms, we get

$\rm \:  =  \: {5z}^{2} - 35z - 3z + 21$

$\rm \:  =  \:5z(z - 7) - 3(z - 7)$

$\rm \:  =  \:(z - 7)(5z - 3)$

On substituting the value of z, we get

$\rm \:  =  \:(x - y - 7)[5(x - y)- 3]$

$\rm \:  =  \:(x - y - 7)[5x - 5y- 3]$

Hence,

$\green{\bf :\longmapsto\: {5(x - y)}^{2} - 38(x - y) + 21}$

$\red{\rm \:  =  \:(x - y - 7)(5x - 5y- 3)}$

❥ Basic Concept 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 given quadratic expression as px + qx and get required factors by grouping the terms.

❥ Explore more :-

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)

Answer 2

Answer:

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

Given expression is

$\rm :\longmapsto\: {5(x - y)}^{2} - 38(x - y) + 21$

Let assume that

$\rm :\longmapsto\:\boxed{ \tt{ \: x - y = z \: }} - - - - (1)$

So, above expression can be rewritten as

$\rm \:  =  \: {5z}^{2} - 38z + 21$

Using splitting of middle terms, we get

$\rm \:  =  \: {5z}^{2} - 35z - 3z + 21$

$\rm \:  =  \:5z(z - 7) - 3(z - 7)$

$\rm \:  =  \:(z - 7)(5z - 3)$

On substituting the value of z, we get

$\rm \:  =  \:(x - y - 7)[5(x - y)- 3]$

$\rm \:  =  \:(x - y - 7)[5x - 5y- 3]$

Hence,

$\green{\bf :\longmapsto\: {5(x - y)}^{2} - 38(x - y) + 21}$

$\red{\rm \:  =  \:(x - y - 7)(5x - 5y- 3)}$

❥ Basic Concept 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 given quadratic expression as px + qx and get required factors by grouping the terms.

❥ Explore more :-

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)