Determine the X-Axis Intersections of the Quadratic Function y = x(x + 1)

To solve the problem of finding the intersection points of the function $y = x(x + 1)$ with the x-axis, follow these steps:

• Step 1: Understand that the function intersects the x-axis where $y = 0$.
• Step 2: Set up the equation $x(x + 1) = 0$.
• Step 3: Solve each part of the product for zero:
• For $x = 0$, the first solution is $x = 0$.
• For $x + 1 = 0$, solving gives us the second solution $x = -1$.

These solutions, $x = 0$ and $x = -1$, correspond to the points $(-1, 0)$ and $(0, 0)$ on the Cartesian plane. Thus, the points of intersection are $(-1, 0)$ and $(0, 0)$.

The correct choice from the provided options is:

• : $(-1, 0), (0, 0)$

Therefore, the solution to the problem is that the function $y = x(x + 1)$ intersects the x-axis at $( -1, 0 ), ( 0, 0 )$.