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#### Albert

##### Well-known member

- Jan 25, 2013

- 1,225

$( a −b)(b − c )(c + a ) = −90$

$( a − b)(b + c)(c − a ) = 42$

$( a + b)(b − c )(c − a ) = −60$

$ find :\,\, a,b,c$

- Thread starter Albert
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- Thread starter
- #1

- Jan 25, 2013

- 1,225

$( a −b)(b − c )(c + a ) = −90$

$( a − b)(b + c)(c − a ) = 42$

$( a + b)(b − c )(c − a ) = −60$

$ find :\,\, a,b,c$

- Mar 31, 2013

- 1,346

a = 3, b= 1, c= 6

$( a −b)(b − c )(c + a ) = −90$

$( a − b)(b + c)(c − a ) = 42$

$( a + b)(b − c )(c − a ) = −60$

$ find :\,\, a,b,c$

for solution

it is not elegent but effecitive

from the ( a − b)(b + c)(c − a ) = 42

as 7 devides RHS and not the other 2 equations we have

b+ c = 7 ( as b and c > 0)

so we get (a-b)(c-a) = 6

so combinations a-b = 6 , c- a = 1

a- b= 3 , c - a = 2

a - b = 2 c - a = 3

a- b = 1 , c- a = 6

and (-6, -1),(-3,-2), (-2,-3) and (-1,6) each can be tried with b+ c to give

a= 3, b = 1 and then they are seen to satisfy other 2 equations

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- #3

- Feb 7, 2012

- 2,785

My solution, like

$( a −b)(b − c )(c + a ) = −90$

$( a − b)(b + c)(c − a ) = 42$

$( a + b)(b − c )(c − a ) = −60$

$ find :\,\, a,b,c$

- Mar 31, 2013

- 1,346

we can combine (from above)My solution, likekaliprasad's, involved a bit of guesswork. Add the first and second equations to get $$-2(a-b)^2c = -48.$$ Add the second and third equations to get $$-2(c-a)^2b = -18.$$ Add the first and third equations to get $$-2(b-c)^2a = -150.$$ You can write those equations as $$(a-b)^2c = 24 = 2^2*6,$$ $$(c-a)^2b = 9 = 3^2*1,$$ $$(b-c)^2a = 75 = 5^2*3.$$ If you then make the assumption that the squared terms on the left side of those equations correspond to the squared terms on the right side, and the nonsquared terms likewise correspond, then you read off the solution $a=3,\,b=1,\,c=6,$ which turns out to be correct.

(a−b)^2c=24=2^2∗6

and b+ c = 7 ( from by solution) and confirm c = 6

we could have (a-b) = 1 and c = 24 from the above and by trial and error remove it.

and combining the two we get c =6 ad confirm and then b= 1 and a = 3

combining the above 2 get less range to guess