A machinist is using a CNC router to carve a hyperbolic design defined by the equation ${}4x^2 - 9y^2 - 24x - 36y - 36 = 0$. After following the procedure to rewrite this in standard form, which coordinates should be used for the center (h, k) of the hyperbola?

A structural engineer is analyzing a hyperbolic support beam defined by the equation ${}4x^2 - 9y^2 - 24x - 36y - 36 = 0$. To determine the beam's geometric properties, the engineer must convert this general equation into its standard form. Arrange the following steps in the correct order to complete the conversion.

A technician is mapping a hyperbolic path defined by the general equation ${}4x^2 - 9y^2 - 24x - 36y - 36 = 0$. After converting the equation to its standard form, $\frac{(x - 3)^2}{9} - \frac{(y + 2)^2}{4} = 1$, the technician identifies the center of the hyperbola at (3, -2). Since the $x^2$ term is positive in the standard form, the technician concludes that the hyperbola opens horizontally and that both vertices will also have a $y$-coordinate of -2. Is this statement true or false?

A structural drafter is converting the general equation of a hyperbolic arch, ${}4x^2 - 9y^2 - 24x - 36y - 36 = 0$, into standard form to determine its physical dimensions. During the grouping step, the drafter factors the terms to yield ${}4(x^2 - 6x) - 9(y^2 + 4y) = 36$. To complete the square for the $y$-terms inside the parentheses, the drafter must add the number ____.

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