A landscaper has 32 square feet of mulch that they want to install in a yard. They need a rectangular shape, with the length to be 4 feet longer than the width, w.

a) Write the equation, in standard form, that would be used to determine the width of the rectangle

(standard form is ax^2+bx+c=0)

b) use the quadratic equation to solve for the width and length of the rectangle

a) is the equation to determine the width of rectangle.

b) Length of rectangle is and width of rectangle is

Step-by-step explanation:

Part a:

Given that the landscaper is rectangular in shape. So to find the equation of width use area formula for rectangle.

Formula for area of rectangle,

To find the length and width, it is given that length is 4 feet longer than width that is,

Also, Area is .

Substituting the values,

Using distributive property and simplifying,

Subtracting 32 from both sides,

Rewriting it in form of ,

So, the equation in standard form which is used to determine the width of rectangle is

Part b:

To solve the above equation use the quadratic formula,

Rewriting it in terms of w,

where, a = 1, b = 8 and c = - 32.

Substituting the values,

Simplifying,

Applying rule,

Hence there are two values of x as follows,

and

and

and

Since value of width cannot be negative so,

Now using to find the length.

Therefore, length and width of rectangle is and respectively.