Question

The area of a rectangle is x2 – 8x + 16. The width of therectangle is x – 4. What is the length of the rectangle?-

Answer 1

To answer this question, we need to remember that the area of a rectangle is given by:

`A_{\text{rectangle}}=l\cdot w`

And we have - from the question - that:

`A_{\text{rectangle}}=x^2-8x+16`

And the width of the rectangle is:

`w=x-4`

If we factor the polynomial that represents the area, we need to find two numbers:

• a * b = 16

,

• a + b = -8

And both numbers are:

• a = -4

,

• b = -4

Since

• -4 * -4 = 16

,

• -4 - 4 = -8

Therefore, we can say that:

`x^2-8x+16=(x-4)(x-4)=(x-4)^2`

Therefore:

`l\cdot w=A_{\text{rectangle}}`
`l=(A_(rec\tan gle))/(w)`

Then the length of the rectangle is:

`l=(x^2-8x+16)/(x-4)=((x-4)(x-4))/(x-4)\Rightarrow(x-4)/(x-4)=1`
`l=((x-4))/((x-4))\cdot(x-4)\Rightarrow l=x-4`

In summary, therefore, the length of the rectangle is x - 4.

`l=x-4`

[We can check this result if we multiply both values as follows:

`A_{\text{rectangle}}=l\cdot w=(x-4)\cdot(x-4)=(x-4)^2_{}`

And we already know that the area of the rectangle is:

`x^2-8x+16=(x-4)^2`

.]