3. \( 2 x^{2}-12 x-65=45 \)

To solve the equation \( 2x^2 - 12x - 65 = 45 \), first, let's simplify by moving all terms to one side: \[ 2x^2 - 12x - 65 - 45 = 0 \] This simplifies to: \[ 2x^2 - 12x - 110 = 0 \] Now, divide every term by 2 to make calculations easier: \[ x^2 - 6x - 55 = 0 \] Next, we can solve this quadratic using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 1 \), \( b = -6 \), and \( c = -55 \). Plugging in the values: \[ x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(-55)}}{2(1)} \] \[ x = \frac{6 \pm \sqrt{36 + 220}}{2} \] \[ x = \frac{6 \pm \sqrt{256}}{2} \] \[ x = \frac{6 \pm 16}{2} \] This gives us two possible solutions: \[ x = \frac{22}{2} = 11 \] and \[ x = \frac{-10}{2} = -5 \] So, the solutions are \( x = 11 \) and \( x = -5 \).