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Home » \[ (x - 8)(x - 4) = 0 \] - Get link 4share

\[ (x - 8)(x - 4) = 0 \] - Get link 4share

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Solution: Apply the **Cauchy-Schwarz Inequality** in the Engel form (also known as Titu’s Lemma):

📌 \frac{x^2}{y} + \frac{y^2}{z} + \frac{z^2}{x} \geq \frac{(x + y + z)^2}{y + z + x} = \frac{3^2}{3} = 3

📌 Equality holds when \( \frac{x}{y} = \frac{y}{z} = \frac{z}{x} \), which implies \( x = y = z \). Given \( x + y + z = 3 \), we get \( x = y = z = 1 \). Substituting:

📌 \frac{1^2}{1} + \frac{1^2}{1} + \frac{1^2}{1} = 1 + 1 + 1 = 3

📌 Thus, the minimum value is \( \boxed{3} \).Question:

📌 A linguist studying the evolution of sign language observes that a triangular gesture board has sides of lengths 13 cm, 14 cm, and 15 cm. Compute the area of the triangle and use it to find the radius of the inscribed circle.

📌 To find the area of the triangle with sides \(a = 13\), \(b = 14\), and \(c = 15\), we use Heron's formula. The semi-perimeter \(s\) is:

📌 The area \(A\) is given by:

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