Question

(2х-1/3)^3 - х(2х-1/3)^2=0

Answer 1

$(2x-\frac{1}{3})^{3} -x(2x-\frac{1}{3})^{2}=0\\ (2x-\frac{1}{3})^2(2x-\frac{1}{3} -x)=0\\ (2x-\frac{1}{3})^2(x-\frac{1}{3})=0\\ (2x-\frac{1}{3})^2=0 \\ x-\frac{1}{3} =0\\x_{1} =\frac{1}{6}\\ x_{2}=\frac{1}{3}$

Answer 2

It seems you've provided an equation that needs to be solved for the variable x:

$(2x - \frac{1}{3})^3 - x(2x - \frac{1}{3})^2 = 0$

Let's work on solving this equation step by step:

1. Factor out common terms: Factor out $(2x - \frac{1}{3})^2$ from both terms:

   $(2x - \frac{1}{3})^2 \cdot (2x - \frac{1}{3}) - x(2x - \frac{1}{3})^2 = 0$

2. Apply the distributive property: Distribute the common factor to both terms:

   $(2x - \frac{1}{3})^2 \cdot (2x - \frac{1}{3} - x(2x - \frac{1}{3})) = 0$

3. Simplify the expression in the parentheses: Perform the multiplication inside the parentheses and simplify:

   $(2x - \frac{1}{3})^2 \cdot (2x - \frac{1}{3} - 2x^2 + \frac{1}{3}x) = 0$

   Now, simplify the expression within the second set of parentheses:

   $(2x - \frac{1}{3})^2 \cdot (-2x^2 + \frac{1}{3}x - \frac{1}{3}) = 0$

4. Expand the square: Expand the square of the binomial term $(2x - \frac{1}{3})^2$:

   $(4x^2 - \frac{4}{3}x + \frac{1}{9}) \cdot (-2x^2 + \frac{1}{3}x - \frac{1}{3}) = 0$

5. Distribute and simplify: Distribute the terms from the first parentheses to the terms in the second parentheses and simplify:

   $-8x^4 + \frac{4}{3}x^3 - \frac{4}{3}x^2 + \frac{2}{3}x^2 - \frac{1}{9}x + \frac{1}{9} = 0$

6. Combine like terms: Combine the like terms:

   $-8x^4 + \frac{4}{3}x^3 - \frac{2}{3}x^2 - \frac{1}{9}x + \frac{1}{9} = 0$

7. Set the equation equal to zero: The equation now is in the form $ax^4 + bx^3 + cx^2 + dx + e = 0$. In order to solve this, you might need to use numerical methods or specialized software since solving quartic equations algebraically can be quite complex.

Keep in mind that solving quartic equations can be challenging and might require the use of numerical methods or specialized software. If you have access to software like Mathematica or numerical libraries in programming languages, those tools can help you find approximate solutions.