Question

The general form of an parabola is 3x²+24x-2y+52=0. What is the standard form of the parabola? Enter your answer by filling in the boxes. Enter any fractions in simplest form.

Answer 1

To convert the given equation, 3x² + 24x - 2y + 52 = 0, into standard form, subtract 3x² + 24x + 52 from both sides, then divide the entire equation by -2 to isolate y and obtain the standard form of the parabola, y = -3/2x² - 12x - 26.

Step-by-step explanation:

The standard form of a parabola is given by the equation y = ax² + bx + c. To convert the given equation, 3x² + 24x - 2y + 52 = 0, into standard form, we need to isolate the y variable on one side by moving the other terms to the right side of the equation.

First, subtract 3x² + 24x + 52 from both sides to get -2y = -3x² - 24x - 52. Then, divide the entire equation by -2 to find y alone, which gives us the standard form of the parabola, y = -3/2x² - 12x - 26.

Answer 2

The standard form of the given parabola is
`\((x + 4)^2 = (2)/(3)(y - 26)\)`.

To convert the general form of a parabola
`\(Ax^2 + Bx + Cy + D = 0\)`into standard form
`\((x - h)^2 = 4p(y - k)\)`, we need to complete the square. The general steps are as follows:

1. Move the
`\(x^2\)` and x terms to one side of the equation.

2. Group the
`\(x^2\)` and x) terms.

3. Complete the square for the
`\(x^2\)` and x terms.

4. Write the equation in the standard form.

Let's apply these steps to your given equation:

`\[3x^2 + 24x - 2y + 52 = 0\]`

1. Move the
`\(x^2\)` and x terms to one side:

`\[3x^2 + 24x = 2y - 52\]`

2. Group the
`\(x^2\) and \(x\)` terms:

`\[3(x^2 + 8x) = 2y - 52\]`

3. Complete the square for the
`\(x^2\) and \(x\)` terms:

`\[3(x^2 + 8x + 16) = 2y - 52 + 3(16)\]`

4. Write the equation in standard form:

`\[3(x + 4)^2 = 2y - 4\]`

Now, to make it in the form
`\((x - h)^2 = 4p(y - k)\)`, we need to divide both sides by 3:

`\[(x + 4)^2 = (2)/(3)(y - 26)\]`