Solve by Factoring x^(2/3)-3x^(1/3)-10=0

Problem

x^(2/3)−3*x^(1/3)−10=0

Solution

1. Substitute a new variable to simplify the quadratic form by letting u=x^(1/3)

2. Rewrite the equation in terms of u noting that u^2=(x^(1/3))^2=x^(2/3)

u^2−3*u−10=0

3. Factor the quadratic trinomial by finding two numbers that multiply to −10 and add to −3

(u−5)*(u+2)=0

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

u−5=0⇒u=5

u+2=0⇒u=−2

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

x^(1/3)=5

x^(1/3)=−2

6. Cube both sides of each equation to isolate x

(x^(1/3))^3=5^3⇒x=125

(x^(1/3))^3=(−2)^3⇒x=−8

Final Answer

x=125,−8

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