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Home » a² + 12² = 13² → a² + 144 = 169 → a² = 25 → a = 5 - Treasure Valley Movers

a² + 12² = 13² → a² + 144 = 169 → a² = 25 → a = 5 - Treasure Valley Movers

📅 April 17, 2026 👤 scraface

Apr 17, 2026

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📌 Compute the sum of the roots of the equation \( v\sqrt{v} - 4v + 5\sqrt{v} - 6 = 0 \) given that all of the roots are positive.

📌 Let's transform the given equation using a substitution. Let \( x = \sqrt{v} \). Then \( v = x^2 \). Substitute into the equation:

📌 x^2 \cdot x - 4x^2 + 5x - 6 = 0

📌 which simplifies to:

📌 x^3 - 4x^2 + 5x - 6 = 0

📌 Now, we need to find the sum of the roots of this cubic equation. In a cubic equation of the form \( ax^3 + bx^2 + cx + d = 0 \), the sum of the roots is given by:

📌 rac{b}{a} = -rac{-4}{1} = 4

📌 Since \( x = \sqrt{v} \), each positive root \( x_i \) corresponds to a positive root \( v_i = x_i^2 \). However, we are asked for the sum of the roots of the original equation in terms of \( v \), not \( x \). The sum of the roots of the original equation in \( v \) corresponds directly to the sum of \( x_i^2 \), but this is not simply the sum of the \( x_i \)'s. Instead, note that since we are only asked for the sum of roots (and given all are positive, and the transformation is valid), the number of valid \( x \)-roots translates to transformable \( v \)-roots, but the sum of the original \( v_i \) values corresponds to the sum of \( x_i^2 \), which is not directly \( 4^2 = 16 \).

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