Two numbers b and c are chosen at random with replacement from the numbers 1 to 9.  Find the probability that x^2 + bx + c > 0 for all x beloging to R

Asked by Shruthi Vidya | 6th Jul, 2014, 08:03: PM

Expert Answer:

I n space o r d e r space t o space h a v e space q u a d r a c t i c space i n e q u a l t i y space p o s i t i v e space f o r space a l l space v a l u e s space o f space x Q E space s h o u l d space b e space i n space f o r m space o f space open parentheses x plus-or-minus k close parentheses squared plus C space w h e r e space C greater than 0 N o w space i n space g i v e n space q u e s t i o n space x squared plus b x plus c greater than 0 I f space w e space t a k e space b equals 9 space t h e n space x squared plus 2 open parentheses 9 over 2 close parentheses x plus open parentheses 9 over 2 close parentheses squared space b u t space c space c a n n o t space t a k e space m o r e space t h a n space 9. space S o space t h i s space c a s e space i s space n o t space p o s s i b l e S i m i l a r l y space f o r space 6 comma space 7 comma space 8 space a n d space 9 space n o space v a l u e space i s space p o s s i b l e space f o r space c  W h e n space w e space t a k e space b equals 5 comma space C space c a n space t a k e space 7 comma space 8 space a n d space 9 space b e c a u s e space space x squared plus 2 open parentheses 5 over 2 close parentheses x plus open parentheses 5 over 2 close parentheses squared space w h e r e space 7 comma space 8 space a n d space 9 space c a n space b e space s p l i t space i n t o space s u m space o f space t w o space n u m b e r s space w i t h space v a l u e s space 6.25 plus k  S i m i l a r l y space f o r space b equals 4 comma space C space c a n space t a k e space 5 comma space 6 comma space 7 comma space 8 comma space 9 b equals 3 comma space C space c a n space t a k e space 3 comma space 4 comma space 5 comma space 6 comma space 7 comma space 8 comma space 9 b equals 2 comma space C space c a n space t a k e space 2 comma space 3 comma space 4 comma space 5 comma space 6 comma space 7 comma space 8 comma space 9 b equals 1 comma space C space c a n space t a k e space a l l space 1 minus 9  S o space n u m b e r space o f space p o s s i b i l i t e s space i n space f a v o u r space a r e space 3 plus 5 plus 7 plus 8 plus 9 equals 32 T o t a l space c h a n c e s space p o s s i b l e equals 9 cross times 9 equals 81  P r o b a b i l i t y equals 32 over 81  space

Answered by Dharma Teja | 7th Jul, 2014, 10:40: AM

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