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Home » The trace is invariant under Lorentz transformations, so: - Leaselab

The trace is invariant under Lorentz transformations, so: - Leaselab

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 x^2 + 1 = 2x \Rightarrow x^2 - 2x + 1 = 0 \Rightarrow (x - 1)^2 = 0 \Rightarrow x = 1

📌 So $ \frac{a + 2b}{a - 2b} = 1 \Rightarrow a + 2b = a - 2b \Rightarrow 4b = 0 \Rightarrow b = 0 $

📌 Now compute $ \frac{a^2 + 4b^2}{a^2 - 4b^2} $. Since $ b = 0 $, this becomes:

📌 \frac{a^2}{a^2} = \boxed{1}

📌 Question: A polynomial $ f(x) $ satisfies $ f(x + 1) - f(x) = 6x + 4 $. Find the leading coefficient of $ f(x) $.

📌 Solution: Since the difference $ f(x+1) - f(x) $ is linear, $ f(x) $ must be a quadratic polynomial. Let $ f(x) = ax^2 + bx + c $. Then:

📌 f(x+1) = a(x+1)^2 + b(x+1) + c = a(x^2 + 2x + 1) + b(x + 1) + c = ax^2 + 2ax + a + bx + b + c

📌 f(x+1) - f(x) = (ax^2 + 2ax + a + bx + b + c) - (ax^2 + bx + c) = 2ax + a + b

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An urban forestry researcher photographs trees in a grid of 4 rows and...
Total number of 1×1 grid positions: \( 4 \times 3 = 12 \).
Number of ways to choose 3 distinct positions: \( \binom{12}{3} = \fra...
Now, favorable outcomes: all 3 positions in different rows.
There are 4 rows, choose 3 distinct rows: \( \binom{4}{3} = 4 \).
Each chosen row has 3 columns, so for each row, pick 1 position: \( 3...
But this counts selections per row order — since we are selecting unor...
So, number of favorable selections:
\binom{4}{3} \times 3^3 = 4 \times 27 = 108
Thus, probability:

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