To transform the given equation of the parabola into its standard form, we follow these steps:

Given equation:

$y^2 + 4x + 14y = -53$

Step 1: Rearrange the terms

First, group the $y$-terms together and move the constant to the other side:

$y^2 + 14y = -4x - 53$

Step 2: Complete the square for the $y$-terms

To complete the square on the left-hand side, look at the coefficient of $y$ (which is 14). Take half of 14 and square it:

$\frac{14}{2} = 7, \quad 7^2 = 49$

Now, add 49 to both sides of the equation:

$y^2 + 14y + 49 = -4x - 53 + 49$

Simplifying:

$(y + 7)^2 = -4x - 4$

Step 3: Simplify the right-hand side

Factor the right-hand side:

$(y + 7)^2 = -4(x + 1)$

Final Standard Form:

The standard form of the equation is:

$(y + 7)^2 = -4(x + 1)$

This is the equation of a parabola that opens to the left.

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Related Questions:

1. How do you determine the vertex of the parabola from the standard form?
2. What is the axis of symmetry for this parabola?
3. How can you find the focus of this parabola?
4. How do you determine whether a parabola opens left, right, up, or down?
5. How would the equation change if the parabola opened upwards?

Tip:

Completing the square is a common technique used to convert quadratic equations into their standard forms.