1. (A) Transform the general equation into standard form, if the given equation of a parabola is y² + 4x 14y= -53

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.

Would you like more details or have any questions?

### 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.

Transform the general equation into standard form, if the given equation of a parabola is y² + 4x + 14y = -53.

Discover how to transform the equation y² + 4x + 14y = -53 into its standard form through completing the square. Learn key algebraic techniques and understand the properties of parabolas with a detailed step-by-step solution. Perfect for students in Grades 9-11.

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