In-equation-ax-2-bx-c-0-a-b-c-are-randomly-selected-from-integers-what-is-the-probability-that-roots-will-be-real-

Question Number 182360 by Rasheed.Sindhi last updated on 08/Dec/22

In equation {ax}^{2} + bx + c = 0, a, b, c are

randomly selected from integers;

what is the probability that roots

will be real ?

Commented by mr W last updated on 08/Dec/22

i^{'} m n o t s u r e i f t h e r e i s a s o l u t i o n .

b u t i t h i n k t h e r e i s a s o l u t i o n w h e n

t h e q u e s t i o n i s e . g . l i k e a, b, c \in R, a n d

0 ⩽ a, b, c ⩽ 10 . i n t h i s c a s e w e h a v e

p = \frac{5 + 6 \ln 2}{36} \approx 25 . 4 %, s e e b e l o w . p l e a s e

c h e c k .

Commented by mr W last updated on 08/Dec/22

t h i s i s a n i c e q u e s t i o n . i h o p e y o u

h a v e a s o l u t i o n f o r a, b, c a s i n t e g e r s .

Commented by Rasheed.Sindhi last updated on 09/Dec/22

N o s i r, I k n o w n o t h i n g a b o u t

a n s w e r / s o l u t i o n . A c t u a l l y {i t}^{'} s b e y o n d

m y c a p a c i t y! B u t o n l y q u e s t i o n

r a i s e d i n m y m i n d a n d I t h r e w

i t f o r d i s c u s s i o n .

Commented by mr W last updated on 09/Dec/22

i s e e, t h a n k s!

i t h i n k w e {c a n}^{'} t a n s w e r t h e q u e s t i o n,

i f w e {d o n}^{'} t r e s t r a i n t h e r a n g e o f t h e

v a l u e s . {t h a t}^{'} s t h e s a m e a s w e {c a n}^{'} t

d e t e r m i n e t h e p r o b a b i l i t y o f d r a w i n g

a b l a c k b a l l f r o m a b a g, i f w e {d o n}^{'} t

k n o w h o w m a n y b a l l s a r e i n t h e b a g

o r i f t h e n u m b e r o f b a l l s i n t h e b a g i s

i n f i n i t e . a n o t h e r i s s u e i s, w h e n

a, b, c a r e i n t e g e r s i n a g i v e n r a n g e,

s a y - 10 ⩽ a ⩽ 20, 10 ⩽ b ⩽ 50, - 20 ⩽ c ⩽ 0

w e c a n s o l v e i t, b u t w e h a v e t o c h e c k

a l l 31 \times 41 \times 21 = 26691 d i s c r e t e p o i n t s

o n e a f t e r o n e t o s e e w h i c h p o i n t s

f u l f i l l b^{2} ⩾ 4 a c a n d g e t t h e n u m b e r

o f s u c h p o i n t s, s a y n_{1} . t h e n w e c a n

c a l c u l a t e t h e p r o b a b i l i t y p = \frac{n_{1}}{26691} .

t h i s i s a n e a s y b u t t o u g h a n d s t u p i d

p r o c e s s .

b u t w h e n a, b, c a r e n o t d i s c r e t e i n t e g e r

v a l u e s, b u t c o n t i n u o u s r e a l v a l u e s i n

a g i v e n r a n g e, w e c a n a p p l y f u n c t i o n s

a n d t h e i r i n t e g r a l s t o f i n d t h e

p r o b a b i l i t y a s i s h o w e d b e l o w .

a n y w a y, t h a n k y o u f o r c r e a t i n g a

v e r y i n t e r e s t i n g q u e s t i o n!

Commented by Rasheed.Sindhi last updated on 09/Dec/22

G rateful sir, for your very deep disscussion

and for your very precious time!

Answered by mr W last updated on 09/Dec/22

s a y t h e q u e s t i o n i s :

a, b, c \in R a n d 0 ⩽ a, b, c ⩽ 10 .

w e k n o w t h e e q u a t i o n {a t}^{2} + b t + c = 0

h a s r e a l r o o t s, w h e n b^{2} ⩾ 4 a c .

l e t a = x, c = y, b = z .

e a c h r a n d o m l y s e l e c t e d v a l u e s f o r

a, b, c r e p r e s e n t a p o i n t i n s i d e t h e c u b e

10 \times 10 \times 10 . t h e e q u a t i o n h a s r e a l

r o o t s, w h e n t h e p o i n t l i e s i n s i d e t h e

z o n e b^{2} ⩾ 4 a c, i . e . z^{2} ⩾ 4 x y . {t h a t}^{'} s t h e

h a t c h e d a r e a s i n t h e d i a g r a m .

t h e p r o b a b i l i t y t h a t t h e r o o t s a r e

r e a l i s t h e n

p = \frac{v o l u m e o f t h e z o n e z^{2} ⩾ 4 x y}{v o l u m e o f t h e c u b e}

Commented by mr W last updated on 08/Dec/22

Commented by mr W last updated on 08/Dec/22

4 x y ⩽ z^{2}

y = \frac{z^{2}}{4 x}

a t y = 10 : x_{1} = \frac{z^{2}}{40}

A (z) = \frac{z^{2}}{40} \times 10 + \int_{x_{1}}^{10} \frac{z^{2}}{4 x} d x

A (z) = \frac{z^{2}}{4} + \frac{z^{2}}{4} \times \ln \frac{400}{z^{2}}

V_{z o n e} = \int_{0}^{10} A (z) d z

V_{z o n e} = \int_{0}^{10} (\frac{z^{2}}{4} + \frac{z^{2}}{4} \times \ln \frac{400}{z^{2}}) d z

V_{z o n e} = \frac{1}{4} (\frac{10^{3}}{3} + \int_{0}^{10} z^{2} \times \ln \frac{400}{z^{2}} d z)

V_{z o n e} = \frac{1000}{4} (\frac{5}{9} + \frac{2 \ln 2}{3}) = \frac{1000 (5 + 6 \ln 2)}{36}

V_{c u b e} = 10^{3} = 1000

\Rightarrow p = \frac{V_{z o n e}}{V_{c u b e}} = \frac{5 + 6 \ln 2}{36} \approx 25 . 4 %

Commented by mr W last updated on 09/Dec/22

i h o p e s o m e b o d y c o u l d c o m f i r m m y

m e t h o d a n d r e s u l t, o r d i s p r o v e i t .

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