Answered

Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Join our platform to connect with experts ready to provide detailed answers to your questions in various areas. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.

Tema III: Factoriza la suma y la diferencia de los siguientes cuadrados.

a) [tex]\(x^2 + 6x + 9\)[/tex]

b) [tex]\(x^2 - 30x + 225\)[/tex]

c) [tex]\(4x^2 + 12x + 9\)[/tex]

d) [tex]\(9x^2 - 12xy + 4y^2\)[/tex]

Sagot :

GinnyAnswer
Vamos a factorizar cada una de las expresiones dadas, paso a paso.

### Parte (a):
[tex]\[ x^2 + 6x + 9 \][/tex]

Este es un trinomio cuadrado perfecto, ya que se ajusta a la forma de [tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex].

Notemos que aquí:
- [tex]\(a = x\)[/tex]
- [tex]\(b = 3\)[/tex]
- [tex]\(2ab = 2(x)(3) = 6x\)[/tex]
- [tex]\(b^2 = 3^2 = 9\)[/tex]

Entonces,
[tex]\[ x^2 + 6x + 9 = (x + 3)^2 \][/tex]

### Parte (b):
[tex]\[ x^2 - 30x + 225 \][/tex]

De nuevo, tenemos un trinomio cuadrado perfecto, de la forma [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex].

Notemos que:
- [tex]\(a = x\)[/tex]
- [tex]\(b = 15\)[/tex]
- [tex]\(2ab = 2(x)(15) = 30x\)[/tex]
- [tex]\(b^2 = 15^2 = 225\)[/tex]

Por lo tanto,
[tex]\[ x^2 - 30x + 225 = (x - 15)^2 \][/tex]

### Parte (c):
[tex]\[ 4x^2 + 12x + 9 \][/tex]

Este también es un trinomio cuadrado perfecto de la forma [tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex].

Aquí:
- [tex]\(a = 2x\)[/tex]
- [tex]\(b = 3\)[/tex]
- [tex]\(2ab = 2(2x)(3) = 12x\)[/tex]
- [tex]\(b^2 = 3^2 = 9\)[/tex]

Entonces,
[tex]\[ 4x^2 + 12x + 9 = (2x + 3)^2 \][/tex]

### Parte (d):
[tex]\[ 9x^2 - 12xy + 4y^2 \][/tex]

Esta expresión es un trinomio cuadrado perfecto de la forma [tex]\((a - b)^2 = a^2 - 2ab + b^2\)[/tex].

En este caso:
- [tex]\(a = 3x\)[/tex]
- [tex]\(b = 2y\)[/tex]
- [tex]\(2ab = 2(3x)(2y) = 12xy\)[/tex]
- [tex]\(b^2 = (2y)^2 = 4y^2\)[/tex]

Por lo tanto,
[tex]\[ 9x^2 - 12xy + 4y^2 = (3x - 2y)^2 \][/tex]

### Resumen de las factorizaciones:
a) [tex]\[ x^2 + 6x + 9 = (x + 3)^2 \][/tex]
b) [tex]\[ x^2 - 30x + 225 = (x - 15)^2 \][/tex]
c) [tex]\[ 4x^2 + 12x + 9 = (2x + 3)^2 \][/tex]
d) [tex]\[ 9x^2 - 12xy + 4y^2 = (3x - 2y)^2 \][/tex]
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.