Solve the Equation: -2x(3-x)+(x-3)² = 9 for Parameter x

To solve the equation $-2x(3-x) + (x-3)^2 = 9$, follow these steps:

• Step 1: Expand each term:
  $-2x(3-x) = -6x + 2x^2$ and $(x-3)^2 = x^2 - 6x + 9$.
• Step 2: Substitute the expanded terms into the equation:
  $-6x + 2x^2 + x^2 - 6x + 9 = 9$.
• Step 3: Combine like terms:
  $(2x^2 + x^2) + (-6x - 6x) + 9 = 9$.
• Step 4: Simplify further:
  $3x^2 - 12x + 9 = 9$.
• Step 5: Move all terms to one side to form a quadratic equation:
  $3x^2 - 12x + 9 - 9 = 0$.
• Step 6: Simplify the expression:
  $3x^2 - 12x = 0$.
• Step 7: Factor out the common term:
  $3x(x - 4) = 0$.
• Step 8: Solve for $x$:
  Since $3x = 0$ or $x - 4 = 0$, we find $x = 0$ or $x = 4$.

Therefore, the values of $x$ that satisfy the equation are $\mathbf{x = 0}$ and $\mathbf{x = 4}$.