Question
Factorize: $(a - b + c)^2 + (b - c + a)^2 + 2(a - b + c)(b - c + a)$
✓
Answer
$(a-b+c)^2+(b-c+a)^2+2(a-b+c)(b-c+a)$
Let $(a-b+c)=x$ and $(b-c+a)=y=x^2+y^2+2 x y$
Using the identity $(a+b)^2=a^2+b^2+2 a b=(x+y)^2$
Now, substituting $x$ and $y(a-b+c+b-c+a)^2$
Cancelling $- b ,+ b \&+ c ,- c =(2 a )^2=4 a ^2$
$\therefore(a-b+c)^2+(b-c+a)^2+2(a-b+c)(b-c+a)=4 a^2$
Let $(a-b+c)=x$ and $(b-c+a)=y=x^2+y^2+2 x y$
Using the identity $(a+b)^2=a^2+b^2+2 a b=(x+y)^2$
Now, substituting $x$ and $y(a-b+c+b-c+a)^2$
Cancelling $- b ,+ b \&+ c ,- c =(2 a )^2=4 a ^2$
$\therefore(a-b+c)^2+(b-c+a)^2+2(a-b+c)(b-c+a)=4 a^2$
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