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Home » "Thor Ragnarok: The Anatomy of the Ultimate Clash—Inside The Legend! - Leaselab

"Thor Ragnarok: The Anatomy of the Ultimate Clash—Inside The Legend! - Leaselab

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 #### \(x = \frac{11}{7}, y = \frac{9}{7}\)Question: Find the number of integer solutions to the equation $ x^2 + y^2 = 25 $, where $ x $ and $ y $ are integers.

📌 Solution: The equation $ x^2 + y^2 = 25 $ represents a circle of radius 5 centered at the origin. We seek all integer pairs $ (x, y) $ that satisfy this. The integer solutions occur when $ x^2 $ and $ y^2 $ are perfect squares summing to 25. The possible square pairs are:

📌 $ (0, 25) $: $ x = 0 $, $ y = \pm5 $ → 2 solutions

📌 $ (9, 16) $: $ x = \pm3 $, $ y = \pm4 $ → 4 solutions (since both $ x $ and $ y $ can independently be positive/negative)

📌 $ (16, 9) $: $ x = \pm4 $, $ y = \pm3 $ → 4 solutions

📌 $ (25, 0) $: $ x = \pm5 $, $ y = 0 $ → 2 solutions

📌 Total solutions: $ 2 + 4 + 4 + 2 = 12 $.

📌 \boxed{12}

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Using the quadratic formula \( t = rac{-b \pm \sqrt{b^2 - 4ac}}{2a} \...
t = rac{10 \pm \sqrt{(-10)^2 - 4 \cdot 9 \cdot 2}}{2 \cdot 9} = rac{...
Simplifying further:
t = rac{10 \pm 2\sqrt{7}}{18} = rac{5 \pm \sqrt{7}}{9}.
To determine which critical point maximizes the growth rate, we evalua...
R''(t) = rac{d}{dt}(9t^2 - 10t + 2) = 18t - 10.
Calculate \( R''(t) \) at each critical point:
For \( t = rac{5 + \sqrt{7}}{9} \):
R''\left(rac{5 + \sqrt{7}}{9}
ight) = 18\left(rac{5 + \sqrt{7}}{9}

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