To rewrite the equation $x^2 - y^2 + 224 - 30x = 0$ in the standard form of a conic, we need to complete the square for the $x$-terms and isolate $y^2$.

Step 1: Group $x$-terms and reorganize

$x^2 - 30x - y^2 + 224 = 0$ Reorganize: $x^2 - 30x - y^2 = -224$

Step 2: Complete the square for $x$

The $x^2 - 30x$ term can be completed by adding and subtracting $\left(\frac{-30}{2}\right)^2 = 225$: $x^2 - 30x + 225 - 225 - y^2 = -224$ $(x - 15)^2 - 225 - y^2 = -224$

Step 3: Simplify

$(x - 15)^2 - y^2 = 225 - 224$ $(x - 15)^2 - y^2 = 1$

Step 4: Identify the conic

The equation is now in the standard form of a hyperbola: $\frac{(x - 15)^2}{1} - \frac{y^2}{1} = 1$

This is a hyperbola centered at $(15, 0)$ with the transverse axis along the $x$-axis.

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Key Features:

1. Center: $(15, 0)$
2. Transverse Axis: Horizontal
3. Vertices: $(15 \pm 1, 0) = (16, 0)$ and $(14, 0)$
4. Asymptotes: $y = \pm (x - 15)$ (from the general form of hyperbolas)
5. Equation Type: Hyperbola.

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Would you like more details or further steps clarified?

Follow-Up Questions:

1. How do you determine the center of a hyperbola?
2. Can you explain the significance of completing the square in conic equations?
3. How do you find the asymptotes of a hyperbola?
4. What are the key differences between hyperbolas and other conic sections?
5. How do the coefficients affect the orientation of the hyperbola?

Tip:

Always check the signs in conic sections' equations to identify whether the conic is a hyperbola, ellipse, or parabola.