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
a.<pre>{{{highlight(highlight(x^(1/3) - 36 = 9x^(1/6)))}}}
{{{highlight(highlight(x^(1/3) - 9x^(1/6) - 36 = 0))}}}

Let {{{highlight(highlight(t = x^(1/6)))}}}
Then {{{t^2 = (x^(1/6))^2}}}, or {{{highlight(highlight(t^(1/3)))}}}
{{{highlight(highlight(x^(1/3) - 9x^(1/6) - 36 = 0))}}} becomes: {{{t^2 - 9t - 36 = 0}}} ------- Resulting quadratic equation after substitution of t for {{{highlight(highlight(x^(1/6)))}}}

b.
Since {{{u = x^4}}}, and u = 16, and u = - 4

{{{u = x^4}}}, with u = 16
{{{16 = x^4}}} 
{{{2^4 = x^4}}} 
Thus, {{{highlight_green(2 = x)}}}

{{{u = x^4}}}, with u = - 4 
{{{- 4 = x^4}}} 
This is an EXTRANEOUS SOLUTION, since NO EXPRESSION raised to a power results in a negative (< 0) value  

Therefore, only solution is: {{{highlight_green(x = 2)}}}