Completing the square

Question

A rectangel is 8 cm longer than it is wide. Its area is 250cm^2.

Find the width of the rectangle.

Find the perimeter of the rectangle.

Show your work and explain the process used for solving this.

Answer 1

Ok, let's remember Area and Perimeter formulas for rectangle:

A = L*W

P = 2L+2W

We are given a relationship between the length and width of the rectangle. The length is 8 cm longer than the width. Write this as an equation.

L = W+8

We are given the area of the rectangle, so let's plug this into the area formula.

250 = L*W

Now, to solve for the width, let's look at the 3 equations we have now:

1) 250 = L*W (Area)

2) L = W+8

3) P = 2L+2W

The 3rd equation involves perimeter, and we're not there yet, so just focus on equations 1 & 2.

Notice that we have 2 equations with 2 variables (L and W). We can solve by substitution.

Plugging in (W+8) in place of L:

250 = (W+8)*W

Solve for W:

250 = W²+8W

Now we have a quadratic equation (0 = W²+8W - 250), so we cannot simply solve algebraically.

We need to complete the square.

Step 1) Divide all terms by the leading coefficient (coefficient of W²), which is 1, so no change.

Step 2) Move constant term to other side of equation by itself.

250 = W²+8W

Step 3) Find the number needed to complete the square by dividing the middle term coefficient (8) in half, and then squaring it. 8/2=4. 4²=16

Step 4) Complete the square by adding this number (16) to both sides of the equation.

250+16=W²+8W+16

Now we can factor the RHS of the equation. It will be a perfect square.

266=(W+4)^2

Now we can use a calculator to take the square root of 266.

16.3095=W+4

W = 12.3095 cm

We found the width of the rectangle! Don't forget units were given as cm.

Now, plug this into the equation with L to solve for the length:

L = W+8

L = (12.3095)+8

L = 20.3095 cm

Double check by multiplying L*W = Area.

20.3095 * 12.3095 = 250

Perfect!

Now find the perimeter:

P = 2L+2W

P = 2(20.3095) + 2(12.3095)

P = 65.238 cm

Remember to write the units, cm.

That's the answer! Hope that helps! :)

Answer 2

First set up an equation using the information you know. You know the area and you have information about its length and width.

Area = 250 cm²

Width = w

Length = 8+w

Formula for area of a rectangle is: Area=Length*Width

Now insert your known information into this formula:

250 = (8+w)*w

Simplify your equation by distributing to get rid of your parentheses:

250=8w+w²

Complete the Square:

-Add in blanks to your equation:

w²+8w+[BLANK]=250+[BLANK]

-Determine what to place in your blank:

BLANK=(b/2)²=(8/2)²=4²=16

-Fill in the blanks

w²+8w+16=250+16

-Factor the left side of your equation and simplify the right side:

(w+4)²=266

Now solve for w by taking the square root of both sides, then subtract 4 from both sides:

w+4 = √266

w+4=16.3

w=12.3 cm

The width of the rectangle is: 12.3 cm

Find the perimeter by finding the length, then adding up all four sides of the rectangle:

L=w+8

L=12.3+8=20.3 cm

Perimeter=L+L+w+w = 20.3+20.3+12.3+12.3 = 65.2 cm

The Perimeter of the rectangle is: 65.2 cm

Answer 3

Let x = width

Let x+8 = length

Area = length x width Plugging in values

250 = x(x+8) Now to solve. Distribute

250 = x²+8x To complete the square, (b/2)² = (8/2)² = 16 Add 16 to both sides of the equation

250+16 = x²+8x+16 Factor

266=(x+4)² Take the square root of both sides. Since this is an application problem, we don't need to worry about the negative value, only the positive one

√266 = √(x+4)² Simplify

√266 = x+4 Subtract 4 from both sides

√266-4 cm = x is the width of the rectangle

The length is √266-4+8 = √266+4

Perimeter = 2W+2L

Perimeter = 2(√266-4)+2(√266+4) Distribute

Perimeter = 2√266-8+2√266+8 Combine like terms

Perimeter = 4√266 cm