Question

Josephine has a rectangular garden with an area of 2x2 + x – 6 square feet

Which expressions can represent the length and width of the garden?

length = x2 – 3 feet; width = 2 feet

length = 2x + 3 feet; width = x – 2 feet

length = 2x + 2 feet; width = x – 3 feet

length = 2x – 3 feet; width = x + 2 feet

Answer 1

factor
2x^2+x-6
trial and error, the factors are the legnth and width
or, we could try multiplying the given options
if we do we find the factored form is
(x+2)(2x-3)
answer is last one

Answer 2

we have

`2x^(2) +x-6`

we know that

The area of a rectangle is equal to

`A=L*W`

where

L is the length side of the rectangle

W is the width side of the rectangle

In this problem

Equate the area to zero and Find the roots of the quadratic equation

`2x^(2) +x-6=0`

Group terms that contain the same variable, and move the constant to the opposite side of the equation

`2x^(2) +x=6`

Factor the leading coefficient

`2(x^(2) +0.5x)=6`

Complete the square. Remember to balance the equation by adding the same constants to each side

`2(x^(2) +0.5x+0.25^(2))=6+0.125`

`2(x^(2) +0.5x+0.0625=6.125`

Rewrite as perfect squares

`2(x+0.25)^(2)=6.125`

`(x+0.25)^(2)=3.0625`

Square root both sides

`(x+0.25)=(+/-)√(3.0625)`

`(x+0.25)=(+/-)1.75`

`x1=-0.25+1.75=1.50`

`x2=-0.25-1.75=-2`

so

`2x^(2) +x-6=2(x+2)(x-1.50)`

`2(x+2)(x-1.50)=(x+2)(2x-3)`

`length=(2x-3)\ feet`

`width=(x+2)\ feet`

therefore

the answer is the option

`length=(2x-3)\ feet` ;
`width=(x+2)\ feet`