1. **Stating the problem:** We need to solve the equation $$2x^2 - 5x + 3 = 0$$ for $x$. 2. **Formula used:** For quadratic equations of the form $$ax^2 + bx + c = 0$$, the solutions are given by the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. **Identify coefficients:** Here, $a=2$, $b=-5$, and $c=3$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-5)^2 - 4 \times 2 \times 3 = 25 - 24 = 1$$ 5. **Evaluate the roots:** Since $\Delta > 0$, there are two real roots: $$x = \frac{-(-5) \pm \sqrt{1}}{2 \times 2} = \frac{5 \pm 1}{4}$$ 6. **Find each root:** - For the plus sign: $$x_1 = \frac{5 + 1}{4} = \frac{6}{4} = 1.5$$ - For the minus sign: $$x_2 = \frac{5 - 1}{4} = \frac{4}{4} = 1$$ 7. **Final answer:** The solutions to the equation are $$x = 1$$ and $$x = 1.5$$. This means the parabola crosses the x-axis at these points.