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B) \( 9 X^{2}-16 y^{2} \)

Ask by Salinas Blake. in Argentina
Nov 11,2024

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La factorización de \( 9X^{2} - 16y^{2} \) es \( (3X - 4y)(3X + 4y) \).

Solución

La expresión \( 9X^{2} - 16y^{2} \) es una diferencia de cuadrados. Se puede factorizar utilizando la identidad \( a^{2} - b^{2} = (a - b)(a + b) \). Primero, identificamos \( a \) y \( b \): - \( a = 3X \) (ya que \( (3X)^{2} = 9X^{2} \)) - \( b = 4y \) (ya que \( (4y)^{2} = 16y^{2} \)) Ahora, aplicamos la identidad de la diferencia de cuadrados: \[ 9X^{2} - 16y^{2} = (3X - 4y)(3X + 4y) \] Por lo tanto, la factorización de \( 9X^{2} - 16y^{2} \) es: \[ (3X - 4y)(3X + 4y) \]

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Did you know that the expression \( 9X^2 - 16Y^2 \) is a type of algebraic structure called a difference of squares? This can be factored into \((3X - 4Y)(3X + 4Y)\), showcasing how two squares can create a difference. This concept is not only essential in algebra but also pops up in geometry when dealing with hyperbolas! In practical terms, factors derived from such expressions can be utilized in various real-world applications like signal processing and optics, where understanding wave behaviors—particularly the relationships and differences between them—can lead to significant advancements. So, next time you come across such expressions, think of all the exciting applications they could relate to!

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