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Home » Calculate the discriminant: \( 1^2 - 4 imes 1 imes (-10100) = 1 + 40400 = 40401 \). - Groen Casting

Calculate the discriminant: \( 1^2 - 4 imes 1 imes (-10100) = 1 + 40400 = 40401 \). - Groen Casting

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 f(y) = 2(y - 3)^2 + 10(y - 3) + 11 = 2(y^2 - 6y + 9) + 10y - 30 + 11 = 2y^2 - 12y + 18 + 10y - 19 = 2y^2 - 2y - 1.

📌 Thus, $ f(x^2 - 3) = 2(x^2 - 3)^2 - 2(x^2 - 3) - 1 = 2(x^4 - 6x^2 + 9) - 2x^2 + 6 - 1 = 2x^4 - 12x^2 + 18 - 2x^2 + 5 = 2x^4 - 14x^2 + 23 $.

📌 Final answer: $\boxed{2x^4 - 14x^2 + 23}$

📌 Question: A comet follows a hyperbolic trajectory described by $ 4x^2 - 9y^2 - 32x + 54y + 115 = 0 $. Determine the center of the hyperbola.

📌 Solution: Rewrite the equation by completing the square. Group $ x $ and $ y $ terms:

📌 4(x^2 - 8x) - 9(y^2 - 6y) = -115.

📌 Complete the square for $ x $: $ x^2 - 8x = (x - 4)^2 - 16 $.

📌 For $ y $: $ y^2 - 6y = (y - 3)^2 - 9 $.

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