Question

Solve for x:
x^2 - x^1/2 =0

Answer 1

x^2 - x^(1/2) = 0
x^(1/2) (x^(3/2) - 1) = 0
this is a difference of cubes, so
√x (√x - 1)(x + √x + 1) = 0
√x = 0
√x = 1
√x = (-1 ±√3 i)/2 ==> x = (-1 ∓ √3 i)/2

Answer 2

OR

x^(1/2) = √x

x^2 - x^(1/2) = 0 is

x² - √x = 0

Add √x to bith sides

x² = √x

Raise both sides to the power of two

x⁴ = x

Subtract x to both sides

x⁴ - x = 0

x ( x³ - 1 ) = 0

1= 1³ so

x ( x³ - 1³ ) = 0

Apply difference of cube formula:

a³ - b³ = ( a - b ) ( a² + a • b + b² )

x³ - 1³ = ( x - 1 ) ( x² + x • 1 + 1² )

x³ - 1³ = ( x - 1 ) ( x² + x + 1 )

Equation x ( x³ - 1³ ) = 0 is:

x ( x - 1 ) ( x² + x + 1 ) = 0

This equation will be equal to zero when:

x = 0

x - 1 = 0

and

x² + x + 1 = 0

The solutions of x² + x + 1 = 0 are:

x = - 1 / 2 - i √3 / 2

and

x = - 1 / 2 + i √3 / 2

So the solutions of x² - √x = 0 are:

x = 0

x = 1

x = - 1 / 2 - i √3 / 2

and

x = - 1 / 2 + i √3 / 2