You are designing a gift card that has a border along one side, as shown. the area of the white part of the card is 30 square centimeters. what is the area of the border?

The area of the white rectangle is length * width and is 30 cm^2.

LW = A

Replace L with r and W with r + 1.

r(r + 1) = 30

Distribute the r.

r^2 + r = 30

Subtract 30 from both sides.

r^2 + r - 30 = 0

Factor the polynomial.
We need two numbers that multiply to -30 and add to 1.
They are 6 and -5.

(r + 6)(r - 5) = 0

r + 6 = 0   or   r - 5 = 0

r = -6   or   r = 5

A dimension of a rectangle (length or width) cannot be a negative number, so we discard the negative answer, r = -6. We keep the answer r = 5.

Now that we know r = 5, the border is r by 2 cm, but r = 5 cm, so the border is 5 cm by 2 cm.

A = LW = 5 cm * 2 cm = 10 cm^2

The area of the border is 10 cm^2.