Let's rewrite the hyperbola equation $-x^2 + 3y^2 - 4x + 6y = 28$ in its standard form step by step:

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Step 1: Organize terms

Group $x$-terms and $y$-terms: $-x^2 - 4x + 3y^2 + 6y = 28$

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Step 2: Factor and prepare for completing the square

1. Factor out $-1$ from the $x$-terms: $-(x^2 + 4x) + 3(y^2 + 2y) = 28$

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Step 3: Complete the square

1. For $x^2 + 4x$:

   • Half of $4$ is $2$, and $2^2 = 4$. Add and subtract $4$ inside the parentheses.

2. For $y^2 + 2y$:

   • Half of $2$ is $1$, and $1^2 = 1$. Add and subtract $1$ inside the parentheses.

Adding these values: $-(x^2 + 4x + 4) + 3(y^2 + 2y + 1) = 28 - (-1)(4) + 3(1)$

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Step 4: Simplify the equation

$-(x + 2)^2 + 3(y + 1)^2 = 28 + 4 + 3$ $-(x + 2)^2 + 3(y + 1)^2 = 35$

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Step 5: Divide through by $35$ to normalize

Divide the entire equation by $35$: $\frac{-(x + 2)^2}{35} + \frac{3(y + 1)^2}{35} = 1$

Reorganize into standard form: $\frac{(x + 2)^2}{35} - \frac{(y + 1)^2}{\frac{35}{3}} = 1$

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Final Standard Form:

$\frac{(x + 2)^2}{35} - \frac{(y + 1)^2}{\frac{35}{3}} = 1$

This represents the hyperbola in its standard form, with:

• Center: $(-2, -1)$
• Transverse axis: Horizontal, because the $x$-term is positive.

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Would you like help visualizing this hyperbola or interpreting its components?

Here are 5 related questions to consider:

1. How do you determine the center of a hyperbola from its equation?
2. What is the difference between a horizontal and vertical hyperbola?
3. How do you find the vertices and foci of a hyperbola?
4. How does completing the square affect the equation's structure?
5. What is the relationship between the transverse and conjugate axes in a hyperbola?

Tip: For hyperbolas, always ensure one squared term is positive and one is negative!