Factories: (a^2-54 )^2-36

Question

Factories:
$\left(a^2-54\right)^2-36$

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Answer

$\left(a^2-54\right)^2-36$
$ =\left(a^2-5 a\right)^2-6^2 $
$ =\left[\left(a^2-5 a\right)-6\right]\left[\left(a^2-5 a\right)+6\right] $
$ =\left(a^2-5 a-6\right)\left(a^2-5 a+6\right)$
In order to factories $a^2-5 a-6$, we will find two number $p$ and $q$ such that $p+q=-5$ and $p q=-6$
Now,
$(-6)+1=-5 \text { And }(-6) \times 1=-6$
Splitting the middle term -5 in the given quadratic as $-6+$ a, we get:
$ a^2-5 a-6=a^2-6 a+a-6 $
$ =\left(a^2-6 a\right)+(a-6) $
$ =a(a-6)+(a-6) $
$ =(a+1)(a-6)$
Now, In order to factories $a^2-5 a+6$, we will find two numbers $p$ and $q$ such that $p+q=-5$ and $p q=6$
Clearly,
$(-2)+(-3)=-5 \text { and }(-2) \times(-3)=6$
Splitting the middle term -5 in the quadratic as $-2 a-3 a$, we get:
$ a^2-5 a+6=a^2-2 a-3 a+6 $
$ =\left(a^2-2 a\right)-(3 a-6) $
$ =a(a-2)-3(a-2) $
$ =(a-3)(a-2) $
$ \therefore\left(a^2-5 a-6\right)\left(a^2-5 a+6\right) $
$ =(a-6)(a+1)(a-3)(a-2) $
$ =(a+1)(a-2)(a-3)(a-6)$

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