Question

Drag each value or expression to the correct location on the equations and sentences. Each value and expression can be used more than once, but not all values and expressions will be used.
Gary works at a bakery, and he needs to select a rectangular cookie sheet with an area of 192 square inches. The area of the cookie sheet is represented by the expression given below.


Complete the given statements, and find the width of the cookie sheet.


x^2+4x

Step 1: x^2+4x=192

Step 2: x(x+__)=192 Length is X. The width is__.

Step 3: x^2+4x-192=0

Step 4: (x+__)(x-12)=0

Because the length can't be__, the length is 12 and the width is __.


Options: 12, 4, x, x+4, -16, 16, x+12, x^2

Answer 1

1) Step 1:given

x² + 4x = 192

2) Step 2: 

x (x + ___ ) = 192. ← you have to find the value that fills in the blank.


To find the value that fills in the first blank, you factor the expression:

x² + 4x = x ( x + 4) ← common factor x.

Then, the second factor, this is the width, is x + 4 ← this is the value to  fill in the first empty box.


And the expression becomes x (x + 4) = 192.
 
The length is x. The width is ______ ← second value to fill in.


Since, the area of a rectangle is length × width, being the length the first factor (x), the width is the other factor (x + 4) ← this goes in the second box.


3) Step 3: given


x² + 4x - 192 = 0


4) (x + ____ ) (x - 12) = 0           ← third empty box


Now, you have to fill in the blank the value that completes the factor.

That value must be such that it times  -12 is - 196, and it - 12 is 4.

That number is 16 ← this is the value that goes in the third box

You you can prove it: 4 × (- 12) = - 196 and 16 - 12 = 4.

Then, the expression becomes: 

(x + 16) (x - 12) = 0

5) Because the length cannot be _____, the length is 12 and the width is __________.

For (x + 16) (x - 12) = 0, either x + 16 = 0 or x - 12 = 0.

x + 16 = 0 ⇒ x = - 16, and x - 12 = 0 ⇒ x = 12. So, since the length cannot be negative (-16) the solution is 12.
⇒
- 16 ← is the value for the fourth box.


And given that you named x the length and x + 4 the width, the width is 12 + 4 = 16 ⇒ 16 is the value that goes in the last box.


Now you can verify: 12 × 16 = 192, which is the area of the rectangle.