# Q. Let a,b,c `in` R. Let there exist two real roots of `ax^2 + bx + c=0` .If `ax^2 + bx + c >0` for all x `in` [-1,1] then `(1 + c/a + |b/a|)` is a) always positive b) always negative c)...

Q. Let a,b,c `in` R. Let there exist two real roots of `ax^2 + bx + c=0` .If `ax^2 + bx + c >0` for all x `in` [-1,1] then `(1 + c/a + |b/a|)` is

a) always positive

b) always negative

c) sometimes positive sometimes negative

d) always zero

My teacher told that answer is (b) but how??

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### 1 Answer

`f(x)=ax^2+bx+c >0 AA x in[-1,1]`

`f(-1)=a-b+c >0`

`-b> -a-c`

`b<a+c` (i)

`f(1)=a+b+c >0`

`b> -a-c` (ii)

from (i) and (ii)

`-a-c < b< a+c`

`|b|<|a+c|`

`|b|<|a||1+c/a|`

`|b|/|a|<|1+c/a|`

`-(1+c/a)<|b|/|a|`

`1+|b|/|a|+c/a>0`

**So answer (a) is true.**