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# Let a, b, c and d be distinct real numbers such that

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Let a, b, c, and d be distinct real numbers such that

a = (sqrt4 + (sqrt5+a))

b = (sqrt4 - (sqrt5+b))

c = (sqrt4 + (sqrt5-c))

d = (sqrt4 - (sqrt5-d))
Compute abcd

Jan 25, 2018
edited by waffles  Jan 25, 2018

#1
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If

$$a=\sqrt4+\sqrt5+a\\ then\\ 0=\sqrt4+\sqrt5\\ 0=2+\sqrt5\\ 2=-\sqrt5$$

This is obviously nonsense.

so

This equation has no real solutions.  In fact I do not think it has any solutions at all.

Perhaps you wrote the question incorrrectly?

Jan 25, 2018
#2
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Sorry, I changed the question

Jan 25, 2018
#3
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There is STILL no solution the way you have the equations written down. However, if you modify the signs, + or - on the first and the fourth, then you have this solution:

Solve the following system:

{a = -a + 2 + sqrt(5) | (equation 1)

b = -b + 2 - sqrt(5) | (equation 2)

c = -c + 2 + sqrt(5) | (equation 3)

d = -d + 2 - sqrt(5) | (equation 4)

Express the system in standard form:

{2 a+0 b+0 c+0 d = 2 + sqrt(5) | (equation 1)

0 a+2 b+0 c+0 d = 2 - sqrt(5) | (equation 2)

0 a+0 b+2 c+0 d = 2 + sqrt(5) | (equation 3)

0 a+0 b+0 c+2 d = 2 - sqrt(5) | (equation 4)

Divide equation 4 by 2:

{2 a+0 b+0 c+0 d = 2 + sqrt(5) | (equation 1)

0 a+2 b+0 c+0 d = 2 - sqrt(5) | (equation 2)

0 a+0 b+2 c+0 d = 2 + sqrt(5) | (equation 3)

0 a+0 b+0 c+d = 1 - (sqrt(5))/(2) | (equation 4)

Divide equation 3 by 2:

{2 a+0 b+0 c+0 d = 2 + sqrt(5) | (equation 1)

0 a+2 b+0 c+0 d = 2 - sqrt(5) | (equation 2)

0 a+0 b+c+0 d = (sqrt(5) + 2)/(2) | (equation 3)

0 a+0 b+0 c+d = 1 - sqrt(5)/2 | (equation 4)

Divide equation 2 by 2:

{2 a+0 b+0 c+0 d = 2 + sqrt(5) | (equation 1)

0 a+b+0 c+0 d = 1 - (sqrt(5))/(2) | (equation 2)

0 a+0 b+c+0 d = 1/2 (2 + sqrt(5)) | (equation 3)

0 a+0 b+0 c+d = 1 - sqrt(5)/2 | (equation 4)

Divide equation 1 by 2:

{a+0 b+0 c+0 d = (sqrt(5) + 2)/(2) | (equation 1)

0 a+b+0 c+0 d = 1 - sqrt(5)/2 | (equation 2)

0 a+0 b+c+0 d = 1/2 (2 + sqrt(5)) | (equation 3)

0 a+0 b+0 c+d = 1 - sqrt(5)/2 | (equation 4)

Collect results:

a = 1/2 (2 + sqrt(5))

b = 1 - sqrt(5)/2

c = 1/2 (2 + sqrt(5))

d = 1 - sqrt(5)/2      If you multiply them together, you get:abcd =1/16

Jan 25, 2018