Question

The general form of an parabola is 3x² 24x−2y 52=0. what is the standard form of the parabola?

Answer 1

To find the standard form of the parabola 3x² + 24x - 2y + 52 = 0, divide by -2, complete the square, and rearrange to get y = -1.5(x + 4)^2 - 2. The vertex is (-4, -2) and the parabola opens downward.

Step-by-step explanation:

To convert the given general form of a parabola 3x² + 24x - 2y + 52 = 0 into the standard form, start by isolating the y-term on one side:

First, divide the entire equation by -2 to simplify.

y = -1.5x² - 12x - 26

Next, complete the square for the x-terms:

1. Add (12/2)^2 = 36 to both sides to complete the square.
2. Rewrite the equation incorporating the completed square and adjust the constant term accordingly.

y = -1.5(x + 4)^2 - 2

This is the standard form of the parabola, where the vertex can easily be determined as (-4, -2) and the parabola opens downward due to the negative coefficient of the x² term.