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16) \( -6 a^{2}-25 a-25 \)

Ask by Bartlett Love. in the United States
Mar 19,2025

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Answer

La factorización del polinomio \( -6a^{2} - 25a - 25 \) es \( -\left(2a + 5\right)\left(3a + 5\right) \).

Solution

1. Escribimos el polinomio: \[ -6a^2 - 25a - 25 \] 2. Extraemos el factor común \(-1\): \[ -6a^2 - 25a - 25 = -\left(6a^2 + 25a + 25\right) \] 3. Factorizamos el trinomio \(6a^2+25a+25\). Primero, calculamos el discriminante: \[ \Delta = 25^2 - 4\cdot6\cdot25 = 625 - 600 = 25 \] Como \(\Delta\) es un cuadrado perfecto, el trinomio se puede factorizar. 4. Buscamos dos números que multiplicados den \(6 \cdot 25 = 150\) y que sumados den \(25\). Estos números son \(10\) y \(15\) (ya que \(10 \cdot 15 = 150\) y \(10 + 15 = 25\)). 5. Reescribimos el término central: \[ 6a^2 + 25a + 25 = 6a^2 + 10a + 15a + 25 \] 6. Agrupamos y factorizamos: \[ \begin{aligned} 6a^2 + 10a + 15a + 25 &= (6a^2 + 10a) + (15a + 25) \\ &= 2a(3a+5) + 5(3a+5) \\ &= (3a+5)(2a+5) \end{aligned} \] 7. Sustituimos en el polinomio original: \[ -6a^2 -25a -25 = -\left(2a+5\right)\left(3a+5\right) \] La factorización final es: \[ -6a^2-25a-25 = -\left(2a+5\right)\left(3a+5\right) \]

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Mind Expander

Let's talk about how we can factor the expression \( -6 a^{2}-25 a-25 \) in a way that's as fun as solving a puzzle! First, we can pull out a negative sign to make things easier: \[ -1 (6a^2 + 25a + 25) \] Now, we need to find two numbers that multiply to \( 6 \times 25 = 150 \) and add to \( 25 \). After testing some combinations, we notice the expression can be factored neatly, leaving us with: \[ -1(3a + 5)(2a + 5) \] So, the final factored form is: \[ -1(3a + 5)(2a + 5) \] This method not only helps simplify polynomials but is also a foundational skill in algebra that you’ll use in various real-world scenarios, like engineering, finance, and even computer science! Understanding how to manipulate these expressions allows you to solve complex problems and model real-life situations efficiently.

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