The area A of a rectangular garden is given by the expression2^2+ 2 − 12. The length L of the garden is given by the expression x2 - 9. Find an expression for the width W of the garden. (Recall that A = LW). What are the excluded values in this calculation and what do they represent in context?

Step-by-step explanation:

The correct question is

The area A of a rectangular garden is given by the expression 2x^2+2x− 12. The length L of the garden is given by the expression x^2 - 9. Find an expression for the width W of the garden. (Recall that A = LW). What are the excluded values in this calculation and what do they represent in context?

we know that

The area of a rectangular garden is given by the formula

we have

substitute

Solve for W

Find the roots of the quadratic equation of the numerator

The formula to solve a quadratic equation of the form

is equal to

in this problem we have

so

substitute in the formula

so

The roots are x=2 and x=-3

The quadratic equation in factored form is equal to

substitute in the above expression of W

Rewrite the denominator as difference of squares

substitute

Remember that

In a quotient, the denominator cannot be equal to zero

so

x=-3 and x=3 are excluded values

x=3 represent a vertical asymptote

x=-3 is not included in the domain of the function because the length cannot be a negative number

Simplify