Question

The area of a rectanglular television screen is 3456 in^2. The width of the screen is 24 inches longer thean the height what is the quadratic equation that represents the area of the screen? What are the diminsions of the screen?

Answer 1

The quadratic equation representing the area of the rectangular television screen is
`x^2 +24x-3456=0`.

Height of the rectangular television screen
`=48` inches, and the width of the rectangular television screen
`= 72` inches.

Explanation:

Given: The area of a rectangular television screen is
`3456\: in^(2)`. The width of the screen is
`24` 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
`x` inches, then width of the rectangular screen be
`(x+24)` inches.

Now, we know that area of a rectangle
`= \text{length}*\text{width}`.

As per question,

`x(x+24)=3456`

`\implies x^(2) +24x=3456`

`\implies x^2+24x-3456=0`

So, the quadratic equation representing the area of the rectangular television screen is
`x^2 +24x-3456=0`.

Now, to find the dimensions, we need to solve
`x^2 +24x-3456=0`.

`x^2 +24x-3456=0`

`\implies x^2 -48x+72x-3456=0`

`\implies x(x-48)+72(x-48)=0`

`\implies (x-48)(x+72)=0`

`\implies x-48=0` or
`x+72=0`

`\implies x=48` or
`-72`

Since,
`x` is the height of the rectangular television screen, and height cannot be negative. So, height of the television is
`48` inches, and width of the television screen is
`48+24=72` inches.

Hence, the quadratic equation representing the area of the rectangular television screen is
`x^2 +24x-3456=0`.

Height of the rectangular television screen
`=48` inches, and the width of the rectangular television screen
`= 72` inches.