the question is solve by completing the square and identify the conic property of 25x(squared)-4y(squared)-72y-424=0

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To complete the square: 25x^2 -4 (y^2 + 18y) - 424 = 0{Pull out the -4 from the y variables}25x^2 -4[(y + 9)^2 - 81] - 424 = 0{Complete the square. 9 comes from half of 18 and 81 comes from half of 18 squared. The number outside of the parenthesis is always negative}25x^2 - 4(y + 9)^2 +324 - 424 = 0{Multiply everything in brackets by -4}25x^2 - 4(y + 9)^2 -100 = 0 {Add like terms}25x^2 - 4(y + 9)^2 = 100{Isolate the variables}[(25x^2)/100] - [4(y + 9)^2/100] = 1{Divide both sides by 100}[(x^2)/4] - [(y + 9)^2]/25 = 1{Simplify}That is the standard equation of a hyperbola. To recognize this, there must be both an x^2 and y^2 term, and a negative sign between them while equal to 1.