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

(x(1/2))3−5=0

3. Apply the formula for the difference of cubes, a3−b3=(a−b)*(a2+a*b+b2) 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

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 5−4*(1)*(25)=−75 is negative, there are no additional real solutions.

Final Answer

x=25

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