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=3^3

(x^(1/2))^3−3^3=0

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

(x^(1/2)−3)*((x^(1/2))^2+3*x^(1/2)+3^2)=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=3^2

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 3^2−4*(1)*(9)=−27 is negative.

Final Answer

x=9

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