Question

the area of a rectangular garden in square is xsquare -5x-300 If x=45 what is the width and the height of the garden 12a.widthheight 12b.please pot it step by step show all your work thank you

Answer 1

If a rectangle has an area of A and sides b and h, then:

`A=b\cdot h`

Solving for the base:

`b=(A)/(h)`

Basically, the sides b and h could have any value provided that b*h=A.

Nevertheless, this problem seems to want from us to factorize the expression:

`x^2-5x-300`

So that each side is a binomial.

Part a)

To factorize that expression, find two numbers so that if they are added up, the sum is equal to -5, and if they are multiplied, the product is equal to -300.

Since the product is negative, one number must be negative. Since the sum is negative, the biggest number should be the negative one.

Consider the factors of 300:

`300=2\cdot2\cdot3\cdot5\cdot5`

Using those factors, we can find pairs of numbers that give 300 as a result from multiplying.

After a bit of trial and error, notice that 15*20=300. If we choose 20 as the negative number, then 15*(-20)=-300 and 15+(-20)=-5. Therefore:

`x^2-5x-300=(x+15)(x-20)`

So, we can choose the width and the height to be those factors. Since (x+15) is greater then (x-20), then:

`\begin{gathered} \text{Width}=x+15 \\ \text{Height}=x-20 \end{gathered}`

Part b)

If x=45, then:

`\begin{gathered} \text{Width}=60feet \\ \text{Height}=25feet \end{gathered}`