Final answer:
To complete the square of the given quadratic equation, we rearrange the terms and add/subtract the square of half the coefficient of y. The vertex is (-20, -16), the axis of symmetry is x = -20, the focus is (20, 240), the directrix is y = -272, the equation opens upwards, and the latus rectum has a length of 1024.
Step-by-step explanation:
To complete the square for the given quadratic equation, x + 3y^2 + 32y + 20, we rearrange the terms to group the x and y terms together. The equation becomes (y^2 + 32y) + (x + 20). To complete the square on the y terms, we need to add and subtract the square of half the coefficient of y, which is (32/2)^2 = 256. We also need to add and subtract the same value inside the parentheses to balance the equation. So, the equation becomes (y^2 + 32y + 256) - 256 + (x + 20).
The vertex form of the equation is now (y + 16)^2 - 256 + (x + 20). The vertex of a quadratic equation in the form (y + k)^2 + h is given by (-h, -k). Hence, the vertex of the given equation is (-20, -16).
The axis of symmetry of a quadratic equation is a vertical line passing through the vertex. Thus, the equation's axis of symmetry is x = -20.
The focus and directrix of a quadratic equation in the form (y + k)^2 + h are given by (h, k + p) and y = k - p, respectively, where p is the distance from the vertex to the focus or directrix. In this equation, p = 256, so the focus is at (20, -16 + 256) = (20, 240) and the directrix is y = -16 - 256 = -272.
The given equation opens upwards because the coefficient of y^2 is positive.
The latus rectum is a line segment passing through the focus and perpendicular to the axis of symmetry. Its length is equal to 4p, where p is the distance from the vertex to the focus. In this case, the latus rectum has a length of 4 * 256 = 1024.