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

1. **Rewrite the equation in standard form:** Start with \[ 2x^{2} - 12x - 65 = 45. \] Subtract 45 from both sides: \[ 2x^{2} - 12x - 65 - 45 = 0, \] which simplifies to: \[ 2x^{2} - 12x - 110 = 0. \] 2. **Simplify the equation:** Divide the entire equation by 2 to ease calculation: \[ x^{2} - 6x - 55 = 0. \] 3. **Solve the quadratic equation using the quadratic formula:** For a quadratic equation of the form \[ ax^{2} + bx + c = 0, \] the solutions are given by \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}. \] Here, \(a = 1\), \(b = -6\), and \(c = -55\). 4. **Calculate the discriminant:** \[ \Delta = b^{2} - 4ac = (-6)^{2} - 4(1)(-55) = 36 + 220 = 256. \] 5. **Find the solutions:** Substitute the values into the quadratic formula: \[ x = \frac{-(-6) \pm \sqrt{256}}{2(1)} = \frac{6 \pm 16}{2}. \] This gives two solutions: - For the positive square root: \[ x = \frac{6 + 16}{2} = \frac{22}{2} = 11. \] - For the negative square root: \[ x = \frac{6 - 16}{2} = \frac{-10}{2} = -5. \] 6. **Final Answer:** The solutions to the equation \(2x^{2} - 12x - 65 = 45\) are \[ x = 11 \quad \text{and} \quad x = -5. \]