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.
