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Home » $\omega^2 = \omega - 1$ - Network4

$\omega^2 = \omega - 1$ - Network4

📅 May 17, 2026 👤 scraface

May 17, 2026

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📌 x = \frac{25}{8} - \frac{23}{16} = \frac{50}{16} - \frac{23}{16} = \frac{27}{16}

📌 $\boxed{\frac{27}{16}}$Question: Compute the value of $ \left( \sum_{k=1}^{6} k^2 \right) \times \left( \prod_{j=1}^{3} (2j + 1) \right) $.

📌 Solution: First, compute the sum $ \sum_{k=1}^{6} k^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 1 + 4 + 9 + 16 + 25 + 36 = 91 $. Next, compute the product $ \prod_{j=1}^{3} (2j + 1) = (2\cdot1 + 1)(2\cdot2 + 1)(2\cdot3 + 1) = 3 \cdot 5 \cdot 7 = 105 $. Now multiply the results: $ 91 \times 105 = 9105 $. Therefore, the value is $ \boxed{9105} $.

📌 Question: Find the $ y $-intercept of the line passing through the points $ (3, 7) $ and $ (9, 19) $.

📌 Solution: The slope $ m $ of the line is $ \frac{19 - 7}{9 - 3} = \frac{12}{6} = 2 $. Using point-slope form with point $ (3, 7) $: $ y - 7 = 2(x - 3) $. Simplify: $ y = 2x - 6 + 7 = 2x + 1 $. The $ y $-intercept occurs when $ x = 0 $, so $ y = 2(0) + 1 = 1 $. Thus, the $ y $-intercept is $ \boxed{1} $.

📌 Question: Factor the expression $ x^4 - 13x^2 + 36 $ completely over the real numbers.

📌 Solution: Let $ u = x^2 $, so the expression becomes $ u^2 - 13u + 36 $. Factor: $ (u - 9)(u - 4) $. Substitute back: $ (x^2 - 9)(x^2 - 4) $. Each is a difference of squares: $ (x - 3)(x + 3)(x - 2)(

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