Answer
$$x=\frac{\sqrt{17}+3}{4},\:x=\frac{-\sqrt{17}+3}{4}$$
Work Step by Step
To find the x-intercepts, we set the equation equal to zero and then complete the square. Doing this, we find:
$$ 2x^2-3x=1\\ x^2-\frac{3x}{2}=\frac{1}{2}\\ x^2-\frac{3x}{2}+\left(-\frac{3}{4}\right)^2=\frac{1}{2}+\left(-\frac{3}{4}\right)^2\\ \left(x-\frac{3}{4}\right)^2=\frac{17}{16}\\ x=\frac{\sqrt{17}+3}{4},\:x=\frac{-\sqrt{17}+3}{4}$$