Given a quadratic function $y$ with respect to $x$.  $y$ takes the values of $-1, -2$ and $1$ for the values of $x=-1,\ 0,\ 1$ respectively.
Find the length of the line segment of the graph of the curve cut by the $x$-axis.

1956 Tokyo Institute of Technology entrance exam
1. **Determine the quadratic function:**
   Given the values of \( y \) for specific \( x \) values, we can set up a system of equations to find the coefficients of the quadratic function \( y = ax^2 + bx + c \).

   - For \( x = -1 \), \( y = -1 \):
     \[
     a(-1)^2 + b(-1) + c = -1 \implies a - b + c = -1 \quad \text{(Equation 1)}
     \]
   - For \( x = 0 \), \( y = -2 \):
     \[
     a(0)^2 + b(0) + c = -2 \implies c = -2 \quad \text{(Equation 2)}
     \]
   - For \( x = 1 \), \( y = 1 \):
     \[
     a(1)^2 + b(1) + c = 1 \implies a + b + c = 1 \quad \text{(Equation 3)}
     \]

2. **Solve the system of equations:**
   Substitute \( c = -2 \) from Equation 2 into Equations 1 and 3:
   - Equation 1:
     \[
     a - b - 2 = -1 \implies a - b = 1 \quad \text{(Equation 4)}
     \]
   - Equation 3:
     \[
     a + b - 2 = 1 \implies a + b = 3 \quad \text{(Equation 5)}
     \]

   Add Equations 4 and 5 to eliminate \( b \):
   \[
   (a - b) + (a + b) = 1 + 3 \implies 2a = 4 \implies a = 2
   \]

   Substitute \( a = 2 \) back into Equation 4:
   \[
   2 - b = 1 \implies b = 1
   \]

   Therefore, the quadratic function is:
   \[
   y = 2x^2 + x - 2
   \]

3. **Find the roots of the quadratic function:**
   To find where the curve cuts the \( x \)-axis, set \( y = 0 \):
   \[
   2x^2 + x - 2 = 0
   \]

   Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
   \[
   a = 2, \quad b = 1, \quad c = -2
   \]
   \[
   x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 2 \cdot (-2)}}{2 \cdot 2} = \frac{-1 \pm \sqrt{1 + 16}}{4} = \frac{-1 \pm \sqrt{17}}{4}
   \]

4. **Calculate the distance between the roots:**
   The roots are \( x_1 = \frac{-1 + \sqrt{17}}{4} \) and \( x_2 = \frac{-1 - \sqrt{17}}{4} \).

   The distance between the roots is:
   \[
   \left| \frac{-1 + \sqrt{17}}{4} - \frac{-1 - \sqrt{17}}{4} \right| = \left| \frac{\sqrt{17} + \sqrt{17}}{4} \right| = \left| \frac{2\sqrt{17}}{4} \right| = \frac{\sqrt{17}}{2}
   \]

The final answer is \(\boxed{\frac{\sqrt{17}}{2}}\).