X2/3-x=2x/3-x

To solve the equation \( \frac{x^2}{3} - x = \frac{2x}{3} - x \), we can follow these steps: 1. First, simplify both sides of the equation. 2. Combine like terms. Starting with the original equation: \[ \frac{x^2}{3} - x = \frac{2x}{3} - x \] Subtract \(-x\) from both sides: \[ \frac{x^2}{3} - x + x = \frac{2x}{3} - x + x \] This simplifies to: \[ \frac{x^2}{3} = \frac{2x}{3} \] Now, multiply both sides by 3 to eliminate the fractions: \[ x^2 = 2x \] Next, move all terms to one side to set the equation to zero: \[ x^2 - 2x = 0 \] Now, factor the left side: \[ x(x - 2) = 0 \] Setting each factor to zero gives us the solutions: 1. \( x = 0 \) 2. \( x - 2 = 0 \) which gives \( x = 2 \) Thus, the solutions to the equation \( \frac{x^2}{3} - x = \frac{2x}{3} - x \) are: \[ \boxed{0 \text{ and } 2} \]