Solve by Factoring x^(3/2)=125

Problem

x^(3/2)=125

Solution

1. Rewrite the equation by moving all terms to one side to set the equation to zero.

x^(3/2)−125=0

2. Recognize the expression as a difference of cubes, since x^(3/2)=(x^(1/2))^3 and 125=5^3

(x^(1/2))^3−5^3=0

3. Apply the formula for the difference of cubes, a^3−b^3=(a−b)*(a^2+a*b+b^2) where a=x^(1/2) and b=5

(x^(1/2)−5)*((x^(1/2))^2+5*x^(1/2)+25)=0

4. Simplify the terms inside the second factor.

(x^(1/2)−5)*(x+5*x^(1/2)+25)=0

5. Set the first factor to zero to find the real solution.

x^(1/2)−5=0

6. Isolate the variable by adding 5 to both sides.

x^(1/2)=5

7. Square both sides to solve for x

(x^(1/2))^2=5^2

x=25

8. Evaluate the second factor x+5*x^(1/2)+25=0 using the quadratic formula for x^(1/2) Since the discriminant 5^2−4*(1)*(25)=−75 is negative, there are no additional real solutions.

Final Answer

x=25

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