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Home » Thus, the conic is **not degenerate**. - Carbonext

Thus, the conic is not degenerate. - Carbonext

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 But working modulo $7$, we can compute the sum directly using cyclicity. First, find the pattern of $3^k \mod 7$:

📌 3^1 &\equiv 3 \mod 7 \\

📌 3^2 &\equiv 9 \equiv 2 \mod 7 \\

📌 3^3 &\equiv 3 \cdot 2 = 6 \mod 7 \\

📌 3^4 &\equiv 3 \cdot 6 = 18 \equiv 4 \mod 7 \\

📌 3^5 &\equiv 3 \cdot 4 = 12 \equiv 5 \mod 7 \\

📌 3^6 &\equiv 3 \cdot 5 = 15 \equiv 1 \mod 7 \\

📌 So the powers of $3 \mod 7$ repeat every 6 terms. The cycle length is $6$.

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Solution 8:
\( f'(x) = 3x^2 - 12x + 9 \).
Setting \( f'(x) = 0 \) for critical points:
3x^2 - 12x + 9 = 0 \Rightarrow x^2 - 4x + 3 = 0 \Rightarrow (x-3)(x-1)...
So \( x = 1 \) and \( x = 3 \) are critical points.
Check \( f(1) = 2 \):
f(1) = 1^3 - 6(1)^2 + 9(1) + c = 1 - 6 + 9 + c = 4 + c
+ c = 2 \Rightarrow c = -2
#### #### -2
Find the radius of a circle if the circumference is equal to the area.

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\sqrt{1200} = \sqrt{100 \cdot 12} = 10\sqrt{12} = 10 \cdot 2\sqrt{3} =...
The altitude $ h $ corresponding to a side $ b $ is given by:
A = rac{1}{2} b h \Rightarrow h = rac{2A}{b}
To find the longest altitude, we use the shortest side, since altitude...
h_{ ext{longest}} = rac{2 \cdot 20\sqrt{3}}{7} = rac{40\sqrt{3}}{7}...
Thus, the length of the longest altitude is $ oxed{rac{40\sqrt{3}}{7...
We begin by analyzing the general quadratic form:
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
Given equation:
4x^2 - 12xy + 9y^2 + 8x - 36y + 20 = 0

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