Solve by Factoring x^(3/2)=8

Problem

x^(3/2)=8

Solution

1. Subtract 8 from both sides to set the equation to zero.

x^(3/2)−8=0

2. Rewrite the terms as cubes to prepare for the difference of cubes formula, noting that x^(3/2)=(x^(1/2))^3 and 8=2^3

(x^(1/2))^3−2^3=0

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

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

4. Simplify the expression inside the parentheses.

(x^(1/2)−2)*(x+2*x^(1/2)+4)=0

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

x^(1/2)−2=0

6. Solve for x by isolating the radical and squaring both sides.

x^(1/2)=2

(√(,x))^2=2^2

x=4

7. Check the second factor for additional real solutions using the discriminant b^2−4*a*c of the quadratic form u^2+2*u+4 where u=x^(1/2)

2^2−4*(1)*(4)=−12

Since the discriminant is negative, there are no further real solutions.

Final Answer

x=4

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