+79 Let a, b, c, and d be distinct real numbers such that
\(\begin{align*} a &= \sqrt{4 + \sqrt{5 + a}}, \\ b &= \sqrt{4 - \sqrt{5 + b}}, \\ c &= \sqrt{4 + \sqrt{5 - c}}, \\ d &= \sqrt{4 - \sqrt{5 - d}}. \end{align*} \)
Compute abcd.
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
