The circle $x^2+y^2+3 x-y+2=0$ cuts an intercept on $X$-axis of length

The equation of the circle is $x^2 + y^2 + 3x - y + 2 = 0$.

To find the intercept on the $X$-axis, set $y = 0$ in the equation of the circle:

$x^2 + 3x + 2 = 0$

Factor the quadratic equation:

$(x + 1)(x + 2) = 0$

The roots are $x_1 = -1$ and $x_2 = -2$. These are the points where the circle intersects the $X$-axis, specifically $(-1, 0)$ and $(-2, 0)$.

The length of the intercept on the $X$-axis is the absolute difference between these $x$-coordinates:

Length $= |x_1 - x_2| = |-1 - (-2)| = |-1 + 2| = 1$.

Answer: 1