+0  
 
0
95
3
avatar+6794 

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

MaxWong  Aug 16, 2017
Sort: 

3+0 Answers

 #1
avatar
0

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 !!!!.

Guest Aug 16, 2017
 #2
avatar+76107 
+2

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  

 

 

 

cool cool cool 

CPhill  Aug 16, 2017
edited by CPhill  Aug 16, 2017
 #3
avatar+353 
0

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  

AsadRehman  Aug 17, 2017

21 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details