4x to the power 2 upon 3 + 5x to the power 1 upon 3 minus 6 = 0
The given equation is:
4x^(2/3) + 5x^(1/3) - 6 = 0
To solve this equation, we can follow these steps:
Step 1: Make a substitution - Let's introduce a new variable to simplify the equation. Let's say y = x^(1/3).
This substitution simplifies the equation to:
4y^2 + 5y - 6 = 0
Step 2: Solve the quadratic equation - Now, we have a quadratic equation in terms of y. We can solve this equation by factoring, completing the square, or by using the quadratic formula.
The equation can be factored as: (4y - 3)(y + 2) = 0
Setting each factor equal to zero gives:
4y - 3 = 0 --> y = 3/4
y + 2 = 0 --> y = -2
Step 3: Substitute back to find the solutions for x - Since we introduced y as a substitute for x^(1/3), we can substitute the values of y back into the equation to find the values of x.
Substituting y = 3/4:
x^(1/3) = 3/4
Raise both sides to the power of 3 to remove the cube root:
(x^(1/3))^3 = (3/4)^3
x = (3/4)^3 = 27/64
Substituting y = -2:
x^(1/3) = -2
Raise both sides to the power of 3:
(x^(1/3))^3 = (-2)^3
x = (-2)^3 = -8
So, the solutions to the equation 4x^(2/3) + 5x^(1/3) - 6 = 0 are x = 27/64 and x = -8.