18(х+6)-18х=0.5х(х+6)

To solve the equation \( 18(x + 6) - 18x = 0.5x(x + 6) \), follow these steps:

1. Expand both sides of the equation.

   Left side: \[ 18(x + 6) - 18x = 18x + 108 - 18x = 108 \]

   Right side: \[ 0.5x(x + 6) = 0.5x^2 + 3x \]

   So, the equation simplifies to: \[ 108 = 0.5x^2 + 3x \]

2. Rearrange the equation to set it to zero: \[ 0.5x^2 + 3x - 108 = 0 \]

3. To eliminate the decimal, multiply the entire equation by 2: \[ x^2 + 6x - 216 = 0 \]

4. Now, you can solve this quadratic equation using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1 \), \( b = 6 \), and \( c = -216 \).

   First, calculate the discriminant: \[ b^2 - 4ac = 6^2 - 4(1)(-216) = 36 + 864 = 900 \]

   Now, apply the quadratic formula: \[ x = \frac{-6 \pm \sqrt{900}}{2(1)} \] \[ x = \frac{-6 \pm 30}{2} \]

5. Solve for the two possible values of \( x \): \[ x = \frac{24}{2} = 12 \quad \text{and} \quad x = \frac{-36}{2} = -18 \]

Thus, the solutions to the equation are: \[ x = 12 \quad \text{and} \quad x = -18 \]