https://file.aiquickdraw.com/mathbot/file/1728289503119nk2v2c8c.jpg

The image contains three quadratic equations, and you are asked to "transform the given equations to standard form (SF)." 

Here are the equations as they appear:

1. \( 5x^2 + 2y^2 + 12y + 8 = 0 \)
2. \( x^2 + 2y^2 + 4x - 4 = 0 \)
3. \( 9x^2 + 14y^2 + 132y + 297 = 0 \)

To transform these equations into standard form, you will likely need to complete the square for both the \(x\) and \(y\) terms. Standard form for conic sections typically looks like:

\[
Ax^2 + By^2 + Cx + Dy + E = 0
\]
or 
\[
(x - h)^2/a^2 + (y - k)^2/b^2 = 1
\]
for ellipses, hyperbolas, or parabolas depending on the equation.

Would you like a detailed step-by-step solution for each equation? 

Here are 5 questions to consider as follow-ups:
1. Do you know which conic section each of these equations represents?
2. Would you like a general refresher on completing the square?
3. Are there specific steps you would like clarified in completing these transformations?
4. Do you want to see the geometric interpretation of these conic sections?
5. Are there other transformations or changes you’d like to apply (e.g., rotations)?

**Tip:** Completing the square is often essential for transforming quadratic equations into standard form for conics.

Transform the given quadratic equations to standard form: 1. 5x^2 + 2y^2 + 12y + 8 = 0, 2. x^2 + 2y^2 + 4x - 4 = 0, 3. 9x^2 + 14y^2 + 132y + 297 = 0

Learn how to transform quadratic equations into standard form for conic sections using completing the square. This problem involves key concepts such as conic sections, algebraic manipulation, and the method of completing the square. Suitable for Grades 10-12.

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