Solve by Factoring x^(2/3)-4x^(1/3)-5=0

Problem

x(2/3)−4*x(1/3)−5=0

Solution

1. Substitute a new variable to simplify the equation. Let u=x(1/3) Then u2=(x(1/3))2=x(2/3)

2. Rewrite the original equation in terms of u to form a standard quadratic equation.

u2−4*u−5=0

3. Factor the quadratic expression by finding two numbers that multiply to −5 and add to −4 These numbers are −5 and 1

(u−5)*(u+1)=0

4. Solve for u by setting each factor equal to zero.

u−5=0⇒u=5

u+1=0⇒u=−1

5. Back-substitute x(1/3) for u to find the values of x

x(1/3)=5

x(1/3)=−1

6. Cube both sides of each equation to isolate x

(x(1/3))3=5⇒x=125

(x(1/3))3=(−1)3⇒x=−1

Final Answer

x=125,−1

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