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kudzordzifrancis
ANSWER

[tex]w = (x - 8)[/tex]

EXPLANATION

The given rectangle has area,

[tex]A = {x}^{2} - x - 72 \: {m}^{2} [/tex]

It was given to us that, the length of the rectangle is l=(x+8)

To find the width, we need to factor, the expression for the area.

We split the middle term to get,

[tex]A = {x}^{2} - 9x + 8x- 72 [/tex]

We now factor by grouping;

[tex]A = x(x - 9) + 8(x- 9)[/tex]

[tex]A = (x + 8)(x - 9)[/tex]

We know that area of a rectangle is

[tex]A = l \times w[/tex]

Hence the width of the rectangle is,

[tex]w = (x - 8)[/tex]
mixter17

Hello!

The answer is:

The expression that represents the width of the rectangle is:

[tex]width=x-9[/tex]

Why?

We know that the area of a rectangle is equal to:

[tex]A=length*width[/tex]

We are given the function that represents the area of the rectangle and one of its sides, the length is equal to (x+8)

So, rewriting we have:

[tex]x^{2}-x-72 =(x+8)*width[/tex]

The coefficients of the variables of the given function are:

[tex]x^{2}=1\\-x=-1[/tex]

Now, to find the width, we need to find two numbers which its product gives as result "-72" and its addition gives as result the coefficient of the linear term of the function (x), its "-1".

Then,

We know that,

[tex](-9)*(8)=-72\\-9+8=-1[/tex]

So, the factors of the given function are:

[tex]x^{2}-x-72 =(x+8)*(x-9)[/tex]

Since, we already know that the length corresponds to "x+8", we know that the width corresponds to "x-9".

Proving that the factors are right,

Applying the distributive property, we have:

[tex](x+8)(x-9)=x^{2}-9x+8x-72=x^{2}-x-72[/tex]

Hence, the equation is satisfied and we can conclude that:

[tex]length=x+8\\width=x-9[/tex]

Have a nice day!

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