Respuesta :
Answer:
The quadratic equation representing the area of the rectangular television screen is [tex]x^2 +24x-3456=0[/tex].
Height of the rectangular television screen [tex]=48[/tex] inches, and the width of the rectangular television screen [tex]= 72[/tex] inches.
Step-by-step explanation:
Given: The area of a rectangular television screen is [tex]3456\: in^{2}[/tex]. The width of the screen is [tex]24[/tex] inches longer than the height.
To find: The quadratic equation that represents the area of the screen, and what are the dimensions of the screen?
Solution:
Let the height of rectangular television screen be [tex]x[/tex] inches, then width of the rectangular screen be [tex](x+24)[/tex] inches.
Now, we know that area of a rectangle [tex]= \text{length}\times\text{width}[/tex].
As per question,
[tex]x(x+24)=3456[/tex]
[tex]\implies x^{2} +24x=3456[/tex]
[tex]\implies x^2+24x-3456=0[/tex]
So, the quadratic equation representing the area of the rectangular television screen is [tex]x^2 +24x-3456=0[/tex].
Now, to find the dimensions, we need to solve [tex]x^2 +24x-3456=0[/tex].
[tex]x^2 +24x-3456=0[/tex]
[tex]\implies x^2 -48x+72x-3456=0[/tex]
[tex]\implies x(x-48)+72(x-48)=0[/tex]
[tex]\implies (x-48)(x+72)=0[/tex]
[tex]\implies x-48=0[/tex] or [tex]x+72=0[/tex]
[tex]\implies x=48[/tex] or [tex]-72[/tex]
Since, [tex]x[/tex] is the height of the rectangular television screen, and height cannot be negative. So, height of the television is [tex]48[/tex] inches, and width of the television screen is [tex]48+24=72[/tex] inches.
Hence, the quadratic equation representing the area of the rectangular television screen is [tex]x^2 +24x-3456=0[/tex].
Height of the rectangular television screen [tex]=48[/tex] inches, and the width of the rectangular television screen [tex]= 72[/tex] inches.
