Three-real-numbers-a-b-c-satisfying-ab-c-10-bc-a-11-ca-b-14-Find-a-b-b-c-c-a-a-1-b-1-c-1-

Question Number 110596 by Aina Samuel Temidayo last updated on 29/Aug/20

Three real numbers a, b, c satisfying

ab + c = 10, bc + a = 11, ca + b = 14 . Find

(a - b) (b - c) (c - a) (a - 1) (b - 1) (c - 1)

Commented by Her_Majesty last updated on 29/Aug/20

o n e s o l u t i o n i s

a = 3 b = 2 c = 4 \Rightarrow a n s w e r i s - 12

Commented by Aina Samuel Temidayo last updated on 29/Aug/20

Thanks but I {don}^{'} t only need answers .

I also need well detailed solutions .

Answered by Her_Majesty last updated on 30/Aug/20

a b + c = 10

b c + a = 11

c a + b = 14

=====

c = 10 - a b = \frac{11 - a}{b} = \frac{14 - b}{a}

10 - a b = \frac{14 - b}{a} \Rightarrow b = \frac{2 (5 a - 7)}{a^{2} - 1}

c = \frac{2 (7 a - 5)}{a^{2} - 1} = - \frac{(a - 11) (a^{2} - 1)}{2 (5 a - 7)}

\frac{2 (7 a - 5)}{a^{2} - 1} = - \frac{(a - 11) (a^{2} - 1)}{2 (5 a - 7)}

a^{5} - 11 a^{4} - 2 a^{3} + 162 a^{2} - 295 a + 129 = 0

t r y i n g f a c t o r s o f 129 w h i c h a r e \pm 1, \pm 3, \pm 43, \pm 129

\Rightarrow a_{1} = 3

\Rightarrow b_{1} = 2, c_{1} = 4

n o w w e m u s t s o l v e t h i s :

a^{4} - 8 a^{3} - 26 a^{2} + 84 a - 43 = 0

s a d l y {t h e r e}^{'} s n o o t h e r e x a c t s o l u t i o n

a_{2} \approx - 4.07488

a_{3} \approx . 685287

a_{4} \approx 1.56780

a_{5} \approx 9.82180

a n y w a y a l l 5 t r i p l e t s (a / b / c) g i v e - 12