Quadratic...

Show that
(x-2) (x-3) (x-4) (x-5)+2 is always positive

6 Answers

let x-2=a

so, (x-2) (x-3) (x-4) (x-5)+2=a(a+1)(a+2)(a+3)+2.....(1)

now a(a+3)(a+1)(a+2)=(a²+3a)(a²+3a+2)
let a²+3a=k
so the eqn. becomes k(k+2)=k²+2k

so putting the value of a²+3a=k in (1)
a(a+1)(a+2)(a+3)+2=k²+2k+2
=(k+1)²+1 which is always positive.

Draw the graph and observe. The graph for the function is always positive. :)

@Arnab- I think you meant x-5=a

Yes. I am sorry. Thank you Sambit.
I meant x-5=a

@ Arnab thanx....

good question. Had a similar doubt.

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