+0  
 
-1
513
3
avatar+644 

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

waffles  Jan 25, 2018
edited by waffles  Jan 25, 2018
 #1
avatar+94126 
+1

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?

Melody  Jan 25, 2018
 #2
avatar+644 
0

Sorry, I changed the question

waffles  Jan 25, 2018
 #3
avatar
+1

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

Guest Jan 25, 2018

4 Online Users

New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.