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# solve the equation and find the roots x^(-4/3)-x^(-2/3)-8=0

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solve the equation and find the roots x^(-4/3)-x^(-2/3)-8=0

Guest Dec 10, 2014

#1
+92505
+10

x^(-4/3)-x^(-2/3)-8=0   let u = x^(-2/3)   then we have

u^2 - u -  8  = 0          add 8 to both sides

u^2 - u    =  8             complete the square on both sides

u^2 -  u  + 1/4  = 8.25        factor

(u - 1/2)^2  = 8.25              take both roots

u - 1/2 = ±√8.25                   add 1/2 to both sides

u = ±√8.25 + 1/2

So u = √8.25 + 1/2   or  u = -√8.25 + 1/2

Then, back-substituting, we have

x^(-2/3) = √8.25 + 1/2    raise both sides to the (-3) power

x*2 = (√8.25 + 1/2)^(-3)

Take both roots

x = ±√((√8.25 + 1/2)^(-3))  = ±(√8.25 + 1/2)^(-3/2) = about ± .161

The other solution, u = -√8.25 + 1/2   leads to a non-real answer

Here's a graph.........https://www.desmos.com/calculator/kkomxbtlbj

CPhill  Dec 10, 2014
#1
+92505
+10

x^(-4/3)-x^(-2/3)-8=0   let u = x^(-2/3)   then we have

u^2 - u -  8  = 0          add 8 to both sides

u^2 - u    =  8             complete the square on both sides

u^2 -  u  + 1/4  = 8.25        factor

(u - 1/2)^2  = 8.25              take both roots

u - 1/2 = ±√8.25                   add 1/2 to both sides

u = ±√8.25 + 1/2

So u = √8.25 + 1/2   or  u = -√8.25 + 1/2

Then, back-substituting, we have

x^(-2/3) = √8.25 + 1/2    raise both sides to the (-3) power

x*2 = (√8.25 + 1/2)^(-3)

Take both roots

x = ±√((√8.25 + 1/2)^(-3))  = ±(√8.25 + 1/2)^(-3/2) = about ± .161

The other solution, u = -√8.25 + 1/2   leads to a non-real answer

Here's a graph.........https://www.desmos.com/calculator/kkomxbtlbj

CPhill  Dec 10, 2014
#2
+94100
+5

That looks like a lot of work Chris - The equation looks interesting :)

Melody  Dec 10, 2014