Question

name
date
just the right border

1. hannah is an aspiring artist who enjoys taking nature photographs with her digital camera. her mother, cheryl, frequently attends estate sales in search of unique decorative items. last month cheryl purchased an antique picture frame that she thinks would be perfect for framing one of hannahs recent photographs. the frame is rather large, so the photo needs to be enlarged. cheryl wants to mat the picture. one of hannahs art books suggests that mats should be designed so that the picture takes up 50% of the area inside the frame and the mat covers the other 50%. the inside of the picture frame is 20 inches by 32 inches. cheryl wants hannah to enlarge, and crop, if necessary, her photograph so that it can be matted with a mat of uniform width and, as recommended, take up 50% of the area inside the mat. see the image below.

a. let x denote the width of the mat for the picture. model this situation with a diagram. write an equation in x that models this situation.
b. put the equation from part a in the standard form (ax^{2}+bx + c = 0). can this equation be solved by factoring (using integers)?
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Explanation:

Step 1: Calcular el área total del marco

El área del marco es \(A = 20\times32=640\) pulgadas cuadradas.

Step 2: Expresar las dimensiones de la foto

Si el ancho del marco es \(x\), las dimensiones de la foto son \((20 - 2x)\) y \((32 - 2x)\).

Step 3: Escribir la ecuación

Como la foto ocupa el \(50\%\) del área del marco, \((20 - 2x)(32 - 2x)=0.5\times640\).

(Parte b):

Step 1: Expandir la ecuación

$$
LATEXBLOCK0
$$

Step 2: Verificar la factorización

Buscamos dos números \(m\) y \(n\) tales que \(m + n=- 26\) y \(m\times n = 80\).
Intentando factores de \(80\): \(80 = 2\times40=4\times20 = 5\times16\).
No hay dos enteros \(m\) y \(n\) que cumplan \(m + n=-26\) y \(m\times n = 80\).

Answer:

a. La ecuación es \((20 - 2x)(32 - 2x)=320\).
b. La ecuación en forma estándar es \(x^{2}-26x + 80=0\). No se puede resolver por factorización usando enteros.