+9673 If
\(a=\sqrt{4+\sqrt{4+a}}\\ b=\sqrt{4+\sqrt{4-b}}\\ c=\sqrt{4-\sqrt{4-c}}\\ d=\sqrt{4-\sqrt{4+d}}\)
Then what is the value of abcd?
My friends says that it is 48... but I just can't do that
Hi Max: I'm sure that there is a SHORTER solution for "a", but my "Mathematica 11" gives this ridiculously long answer. Sorry about that.
Solve for a:
a = sqrt(sqrt(a + 4) + 4)
a = sqrt(sqrt(a + 4) + 4) is equivalent to sqrt(sqrt(a + 4) + 4) = a:
sqrt(sqrt(a + 4) + 4) = a
Raise both sides to the power of two:
sqrt(a + 4) + 4 = a^2
Subtract 4 from both sides:
sqrt(a + 4) = a^2 - 4
Raise both sides to the power of two:
a + 4 = (a^2 - 4)^2
Expand out terms of the right hand side:
a + 4 = a^4 - 8 a^2 + 16
Subtract a^4 - 8 a^2 + 16 from both sides:
-a^4 + 8 a^2 + a - 12 = 0
The left hand side factors into a product with three terms:
-(a^2 - a - 4) (a^2 + a - 3) = 0
Multiply both sides by -1:
(a^2 - a - 4) (a^2 + a - 3) = 0
Split into two equations:
a^2 - a - 4 = 0 or a^2 + a - 3 = 0
Add 4 to both sides:
a^2 - a = 4 or a^2 + a - 3 = 0
Add 1/4 to both sides:
a^2 - a + 1/4 = 17/4 or a^2 + a - 3 = 0
Write the left hand side as a square:
(a - 1/2)^2 = 17/4 or a^2 + a - 3 = 0
Take the square root of both sides:
a - 1/2 = sqrt(17)/2 or a - 1/2 = -sqrt(17)/2 or a^2 + a - 3 = 0
Add 1/2 to both sides:
a = 1/2 + sqrt(17)/2 or a - 1/2 = -sqrt(17)/2 or a^2 + a - 3 = 0
Add 1/2 to both sides:
a = 1/2 + sqrt(17)/2 or a = 1/2 - sqrt(17)/2 or a^2 + a - 3 = 0
Add 3 to both sides:
a = 1/2 + sqrt(17)/2 or a = 1/2 - sqrt(17)/2 or a^2 + a = 3
Add 1/4 to both sides:
a = 1/2 + sqrt(17)/2 or a = 1/2 - sqrt(17)/2 or a^2 + a + 1/4 = 13/4
Write the left hand side as a square:
a = 1/2 + sqrt(17)/2 or a = 1/2 - sqrt(17)/2 or (a + 1/2)^2 = 13/4
Take the square root of both sides:
a = 1/2 + sqrt(17)/2 or a = 1/2 - sqrt(17)/2 or a + 1/2 = sqrt(13)/2 or a + 1/2 = -sqrt(13)/2
Subtract 1/2 from both sides:
a = 1/2 + sqrt(17)/2 or a = 1/2 - sqrt(17)/2 or a = sqrt(13)/2 - 1/2 or a + 1/2 = -sqrt(13)/2
Subtract 1/2 from both sides:
a = 1/2 + sqrt(17)/2 or a = 1/2 - sqrt(17)/2 or a = sqrt(13)/2 - 1/2 or a = -1/2 - sqrt(13)/2
a ⇒ -1/2 - sqrt(13)/2 = 1/2 (-1 - sqrt(13)) ≈ -2.30278
sqrt(sqrt(a + 4) + 4) ⇒ sqrt(sqrt((-1/2 - (sqrt(13))/(2)) + 4) + 4) = sqrt(sqrt(7/2 - (sqrt(13))/(2)) + 4) ≈ 2.30278:
So this solution is incorrect
a ⇒ sqrt(13)/2 - 1/2 = 1/2 (sqrt(13) - 1) ≈ 1.30278
sqrt(sqrt(a + 4) + 4) ⇒ sqrt(sqrt(((sqrt(13))/(2) - 1/2) + 4) + 4) = sqrt(sqrt(7/2 + (sqrt(13))/(2)) + 4) ≈ 2.51053:
So this solution is incorrect
a ⇒ 1/2 - sqrt(17)/2 = 1/2 (1 - sqrt(17)) ≈ -1.56155
sqrt(sqrt(a + 4) + 4) ⇒ sqrt(sqrt((1/2 - (sqrt(17))/(2)) + 4) + 4) = sqrt(sqrt(9/2 - (sqrt(17))/(2)) + 4) ≈ 2.35829:
So this solution is incorrect
a ⇒ 1/2 + sqrt(17)/2 = 1/2 (1 + sqrt(17)) ≈ 2.56155
sqrt(sqrt(a + 4) + 4) ⇒ sqrt(sqrt((1/2 + (sqrt(17))/(2)) + 4) + 4) = sqrt(sqrt((sqrt(17) + 9)/(2)) + 4) ≈ 2.56155:
So this solution is correct
The solution is:
Answer: | a = 1/2 + sqrt(17)/2 =2.5615528........
b = b = 1/2 (1 + sqrt(13)) = 2.30277564......
c=c = 1/2 (sqrt(17) - 1) = 1.561552813.....
d =d = 1/2 (sqrt(13) - 1) = 1.30277564........
abcd = ~ 12 The product of these 4 variables certainly does not equal 48 but 12 !!!!.
+129852 a = sqrt [ 4 + sqrt (4 + a) ] square both sides
a^2 = 4 + sqrt ( 4 + a)
a^2 - 4 = sqrt (4 + a) square both sides again
a^4 - 8a^2 + 16 = 4 + a
a^4 - 8a^2 - a + 12 = 0
There are 4 possible real solutions to this
The only correct one is ... a = [ 1 + sqrt (17 ] / 2
Likewise...setting each of the other equations up as 4th power polynomials by squaring each side twice produces
b = [ 1 + sqrt (13 ) ] / 2
c = [-1 + sqrt(17) ] / 2
d = [ -1 + sqrt (13) ] / 2
So....abcd = acbd =
[ 1 + sqrt (17)] / 2 * [-1 + sqrt(17) ] /2 * [ 1 + sqrt (13 ) ] / 2 * [ -1 + sqrt (13) ] / 2 =
[ -1 + 17 ] / 4 * [ -1 + 13] / 4 =
[ 16 / 4 ] * [ 12 / 4 ] =
4 * 3 =
12
+476 a = sqrt [ 4 + sqrt (4 + a) ] square both sides
a^2 = 4 + sqrt ( 4 + a)
a^2 - 4 = sqrt (4 + a) square both sides again
a^4 - 8a^2 + 16 = 4 + a
a^4 - 8a^2 - a + 12 = 0
There are 4 possible real solutions to this
The only correct one is ... a = [ 1 + sqrt (17 ] / 2
Likewise...setting each of the other equations up as 4th power polynomials by squaring each side twice produces
b = [ 1 + sqrt (13 ) ] / 2
c = [-1 + sqrt(17) ] / 2
d = [ -1 + sqrt (13) ] / 2
So....abcd = acbd =
[ 1 + sqrt (17)] / 2 * [-1 + sqrt(17) ] /2 * [ 1 + sqrt (13 ) ] / 2 * [ -1 + sqrt (13) ] / 2 =
[ -1 + 17 ] / 4 * [ -1 + 13] / 4 =
[ 16 / 4 ] * [ 12 / 4 ] =
4 * 3 =
12
