find the exact solutions to each of the following equations

Question

find the exact solutions to each of the following equations

 

1. 4x^2/3-4x^1/3=3

2.9x^2/3+4x^-2/3=37

3.3x^2/3-x^1/3=2

 

4. I/2x^2-x=1/3

Answer 1

1. 4x^2/3 - 4x^1/3 - 3 = 0

 

    4(x^1/3)² - 4(x^1/3) - 3 = 0

 

    Let u = x^1/3, 4u² - 4u - 3 = 0

 

                       (2u-3)(2u+1) = 0

 

                       u = 3/2 or u = -1/2

 

But, u = x^1/3, so x^1/3 = 3/2 or x^1/3 = -1/2

 

If x^1/3 = 3/2, then x = (3/2)³ = 27/8

 

If x^1/3 = -1/2, then x = (-1/2)³ = -1/8

 

Solution set {27/8, -1/8}

 

2. 9x^2/3 + 4x^-2/3 = 37

 

   Let u = x^1/3, then 9u² + 4/u² = 37

 

  Multiply the equation by u² to obtain 9u⁴ - 37u² + 4 = 0

 

                                                                (9u² - 1)(u² - 4) = 0

 

                                                                          u² = 1/9, 4 

 

                                                              u = ±1/3, ±2

 

So, x^1/3 = ± 1/3 or ±2

 

        x = ±1/27 or ±8

 

3. Proceed as in #1

 

4. Multiply the equation by 6 to get 3x² -6x - 2 = 0

 

    Solve by the quadratic formula.