Question

factorize: (x-5)²-5(x-5)-24​

Answer 1

Answer:

given:-

$x { }^{2} - 25 - 5x + 25 - 24$

$x {}^{2} - 5x - 24$

hence proved:-It is quadratic equation.

Answer 2

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 quadratic as px + qx and get required factors by grouping the terms.

❥ Solution :-

Given expression is

$\rm :\longmapsto\: {(x - 5)}^{2} - 5(x - 5) - 24$

$\: \: \: \: \: \: \red{ \boxed{ \bf \: Let \: x \: - \: 5 \: = \: y}}$

So, we get

$\rm :\longmapsto\: {y}^{2} - 5y - 24$

Now we have to find p and q in such a way that

• pq = - 24

and

• p + q = - 5

Now,

• - 24 can be factorized as - 8 × 3

and

• - 8 + 3 = - 5

So, p = - 8 and q = 3

Hence,

The above expression after splitting of middle terms can be rewritten as

$\sf \: = \: \: {y}^{2} - 8y + 3y - 24$

$\sf \: = \: \: y(y - 8) + 3(y - 8)$

$\sf \: = \: \: (y - 8)(y + 3)$

Now substituting the value of y = x - 5, we get

$\sf \: = \: \: (x - 5 - 8)(x - 5 + 3)$

$\sf \: = \: \: (x - 13)(x - 2)$