Algebra Question

[crackthegmat2011]

Is x^2 - 14x + (50 + c) always positive, provided x being real ?

1) c is a natural number
2) c is an integer

[ankur.agrawal]

1. C = Natural number , x will never be real as the term inside the discriminant is always negative. (Problem says Provided x being real) . So is it sufficient/insufficient. Confused.

2. C = integer . x can be real / complex ( c=1/-1). Again Confused.

Help.

Is x^2 - 14x + (50 + c) always positive, provided x being real ?

1) c is a natural number
2) c is an integer

[stormier]

Is x^2 - 14x + (50 + c) always positive, provided x being real ?

1) c is a natural number
2) c is an integer

x^2 -14x +50 +c = x^2-14x + 49 + 1 + c = (x-7)^2 +1+c

(x-7)^2 is always positive, if x is real. [it can be negative if x is imaginary; for example if x = 7+2i]

Thus for the above expression to be always positive (1+c) must be greater than zero.


(i) natural number is referred to as either a set of positive integers (1,2,3....) or a set of non negative integers (0,1,2....). By either definition (1+c) > 0. => Sufficient

(ii) c is an integer => c could either be positive or negative. Thus (1+c) can either be negative or positive. => not sufficient

Answer -> A

[Pilot]

x^2-14x+50+c=(x-7)^2+1+c If c>-1 then this expression is always positive.

I is sufficient. because c is natural.
II is not sufficient. Because there infinitely many integers that c<-1.

The answer is A