1. **State the problem:** Solve the quadratic equation $2x^2 - 4x = 3$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$2x^2 - 4x - 3 = 0$$ 3. **Identify coefficients:** Here, $a=2$, $b=-4$, and $c=-3$. 4. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 5. **Calculate the discriminant:** $$b^2 - 4ac = (-4)^2 - 4 \times 2 \times (-3) = 16 + 24 = 40$$ 6. **Substitute values into the formula:** $$x = \frac{-(-4) \pm \sqrt{40}}{2 \times 2} = \frac{4 \pm \sqrt{40}}{4}$$ 7. **Simplify the square root:** $$\sqrt{40} = \sqrt{4 \times 10} = 2\sqrt{10}$$ 8. **Rewrite the solutions:** $$x = \frac{4 \pm 2\sqrt{10}}{4}$$ 9. **Simplify the fraction by dividing numerator and denominator by 2:** $$x = \frac{\cancel{2} \times 2 \pm \cancel{2} \sqrt{10}}{\cancel{2} \times 2} = \frac{2 \pm \sqrt{10}}{2}$$ 10. **Final answer:** $$x = \frac{2 + \sqrt{10}}{2} \quad \text{or} \quad x = \frac{2 - \sqrt{10}}{2}$$