Solve by Factoring x^(3/2)=27

Problem

x(3/2)=27

Solution

1. Subtract 27 from both sides to set the equation to zero.

x(3/2)−27=0

2. Rewrite the terms as cubes to identify a difference of cubes pattern, noting that x(3/2)=(x(1/2))3 and 27=3

(x(1/2))3−3=0

3. Apply the difference of cubes formula, a3−b3=(a−b)*(a2+a*b+b2) where a=x(1/2) and b=3

(x(1/2)−3)*((x(1/2))2+3*x(1/2)+3)=0

4. Simplify the terms inside the second factor.

(x(1/2)−3)*(x+3*x(1/2)+9)=0

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

x(1/2)−3=0

6. Isolate the variable by adding 3 to both sides.

x(1/2)=3

7. Square both sides to solve for x

(x(1/2))2=3

x=9

8. Check the second factor for additional solutions. The quadratic x+3*x(1/2)+9=0 has no real solutions because its discriminant 3−4*(1)*(9)=−27 is negative.

Final Answer

x=9

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