# A+b=8 ab+c+d=23 ad+bc=28cd=12 solve in real numbers the system of equation these points are gifts for those who solve it

1
by dheeerajbolisetti
From trial and error methods, I found that (a,b,c,d)=(4,4,3,4) satisfy the given equations.
asking real numbers. do u want solutions in real numbers or only in integers ?
ur trial error hasi only one solution
real number with cases

2014-12-24T23:02:36+05:30

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a + b = 8  -- equation 1
a b + c + d = 23  -- equation 2
a d + b c = 28    -- equation 3
c d = 12  ---  equation 4

From equation 1 and 4, we get

b = (8 - a)  -- equation 5
c = 12/d    -- eq 6

Substitute these two values in  equation 3
a d + (8 - a) (12/d) = 28
12 a + 28 d - a d^2 - 96 = 0

a = 4 (24 - 7 d) / (12 - d^2)    --- eq  7
Substitute this in equation 5 to get,
b = 4 d (7 - 2 d) / ( 12  - d^2)    -- eq 8

Substitute values from equations 7, 8 and 4 in equation 2.  We get

16d (24 - 7d) (7 - 2d) / (12 - d^2)^2 + 12/d + d = 23
16d^2 (168 - 97 d + 14 d^2) + (12 - d^2)^2 (12 -23 d + d^2) = 0
Simplifying further,
P(d) = d^6 - 23 d^5 + 212 d^4 - 1000 d^3 + 2544 d^2 - 3312 d + 1728 = 0  -- eq 9
There are six real roots for this equation.  They are all positive - can be found found from Descarte's rule for zeroes.

Roots are factors of  12^3 or 1728 ie.,  from the list of 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, etc.

Trying 2, 3, 4, and 6. They are satisfied.  Now, we can write the 6th degree polynomial in d as follows.  here x is not known yet.

P(d) = (d - 2) (d - 3) (d - 4) (d - 6) [d^2 - x d + 1728/(2*3*4*6) ] = 0

Comparing the coefficient of d^3, -23 = -x - 10 -5    => x = 8

factors of (x^2 - 8 d + 1728/144)  are  (d - 2) and (d - 6)

So the roots of P(d) are  d = 2, 3, 4, and 6.  Here  2 and 6 occur twice.

Now for solving the given system of equations.  Given equations are symmetric in a and b.  They are also symmetric in c and d.  Hence there will be solutions with interchange of values of a and b,  as well as  c and d.

Solution 1:
d = 2,  c  = 6 ,  a = (96-56)/(12-4) = 5,  b = 8 - 5 = 3
Solution 2
d = 3,  c = 4,    a = (96 - 84)/(12 - 9) = 4,  b = 8 - 4 = 4
Solution 3
d = 4,  c = 3,  a = (96 - 112)/ (12 - 16) = 4 ,  b = 4
Solution 4
d = 6,  c = 2,  a = 3 ,    b = 8 - 3 = 5

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solutions are :  (5, 3, 6, 2)  ,  (4, 4, 3, 4),  (4, 4, 4, 3)  and  (3, 5, 2, 6)

no ways
what is meant by "no ways ?"
IT WENT ABOVE MY HEAD
NOT THE ANSWER BUT THE QUESTION
u r welcome