SOLUTION: a. Jull wants to make a substitution in the equation: x^(1/3)-36=9x^(1/6) in order to convert it to a quadratic equation. Give the substitution (in form t=...) and the resulting qu

Algebra ->  Equations -> SOLUTION: a. Jull wants to make a substitution in the equation: x^(1/3)-36=9x^(1/6) in order to convert it to a quadratic equation. Give the substitution (in form t=...) and the resulting qu      Log On


   

Question 947349: a. Jull wants to make a substitution in the equation: x^(1/3)-36=9x^(1/6) in order to convert it to a quadratic equation. Give the substitution (in form t=...) and the resulting quadratic equation. Do not solve.
b. After making the substitution u=x^4 in the equation, Jaime found the solution u=16 and u=-4. Find the solution(s) for x.

Thank you
Found 2 solutions by josgarithmetic, MathTherapy:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E%281%2F6%29%2Ax%5E%281%2F6%29
x%5E%281%2F6%2B1%2F6%29
x%5E%282%2F6%29
x%5E%281%2F3%29

That shown above is not rendering to display well on my view.

Use this substitution:
u=x%5E%281%2F6%29 and this gives you u%5E2=x%5E%281%2F3%29.

u=x^(1/6) and u^2=x^(1/3)


Jaime did not use the best choice for substitution.
Also remember, you are first solving for x^(1/6).

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

a. Jull wants to make a substitution in the equation: x^(1/3)-36=9x^(1/6) in order to convert it to a quadratic equation. Give the substitution (in form t=...) and the resulting quadratic equation. Do not solve.
b. After making the substitution u=x^4 in the equation, Jaime found the solution u=16 and u=-4. Find the solution(s) for x.
Thank you
a.
highlight%28highlight%28x%5E%281%2F3%29+-+36+=+9x%5E%281%2F6%29%29%29

highlight%28highlight%28x%5E%281%2F3%29+-+9x%5E%281%2F6%29+-+36+=+0%29%29



Let highlight%28highlight%28t+=+x%5E%281%2F6%29%29%29

Then t%5E2+=+%28x%5E%281%2F6%29%29%5E2, or highlight%28highlight%28t%5E%281%2F3%29%29%29

highlight%28highlight%28x%5E%281%2F3%29+-+9x%5E%281%2F6%29+-+36+=+0%29%29 becomes: t%5E2+-+9t+-+36+=+0 ------- Resulting quadratic equation after substitution of t for highlight%28highlight%28x%5E%281%2F6%29%29%29



b.

Since u+=+x%5E4, and u = 16, and u = - 4



u+=+x%5E4, with u = 16

16+=+x%5E4 

2%5E4+=+x%5E4 

Thus, highlight_green%282+=+x%29



u+=+x%5E4, with u = - 4 

-+4+=+x%5E4 

This is an EXTRANEOUS SOLUTION, since NO EXPRESSION raised to a power results in a negative (< 0) value  



Therefore, only solution is: highlight_green%28x+=+2%29