Solve for x x^(-2/5)-3x^(-1/5)+1=0

Problem

x^(−2/5)−3*x^(−1/5)+1=0

Solution

1. Identify the equation as being in quadratic form by noticing that x^(−2/5) is the square of x^(−1/5)

2. Substitute a new variable u=x^(−1/5) to transform the equation into a standard quadratic.

u^2−3*u+1=0

3. Apply the quadratic formula u=(−b±√(,b^2−4*a*c))/(2*a) to solve for u

u=(−(−3)±√(,(−3)^2−4*(1)*(1)))/(2*(1))

4. Simplify the expression for u

u=(3±√(,9−4))/2

u=(3±√(,5))/2

5. Back-substitute x^(−1/5) for u to solve for x

x^(−1/5)=(3±√(,5))/2

6. Isolate x by raising both sides of the equation to the power of −5

x=((3±√(,5))/2)^(−5)

7. Simplify the negative exponent by taking the reciprocal of the fraction.

x=(2/(3±√(,5)))^5

Final Answer

x=(2/(3±√(,5)))^5

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