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Factoring1

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User Generated
Subject
Algebra
School
Gateway Community College
Type
Homework
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Denote the “another” integer as then the first is   Their product is equal
to 󰇛 󰇜 and is equal to  which given an equation for 
󰇛

󰇜

From it we obtain 
 
󰇛


󰇜
󰇛
 
󰇜


󰇛
󰇜

󰇛
󰇜

󰇛
 
󰇜󰇛
󰇜

This means that or    The first option gives a solution  the
second has no integer solutions. This way,   
Denote the width as then the length is equal to and the area is

󰇛
󰇜

From this,
  and it can be factored as
󰇛

󰇜
󰇛
 
󰇜

󰇛

󰇜

󰇛

󰇜

󰇛

󰇜󰇛

󰇜

This means that  or   The first option gives a solution 
the second has only negative solution  which is impossible. This way,
 
Denote the width as then the length is equal to and the area is

󰇛
󰇜
while the perimeter is
󰇛
󰇜
  This gives an equation
󰇛
󰇜
󰇛

󰇜

or
 
It can be factored as
󰇛

󰇜
󰇛
 
󰇜

󰇛
󰇜
󰇛
󰇜

󰇛
󰇜󰇛
󰇜

This means that or The first option gives a negative solution
 which is impossible, the second gives  This way,
 

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Denote the “another” integer as 𝑛, then the first is 𝑚 = 5𝑛 − 3. Their product is equal to 𝑛(5𝑛 − 3) and is equal to 36, which given an equation for 𝑛, 𝑛(5𝑛 − 3) = 36. From it we obtain 5𝑛2 − 3𝑛 − 36 = 0, (5𝑛2 − 15𝑛) + (12𝑛 − 36) = 0, 5𝑛(𝑛 − 3) + 12(𝑛 − 3) = 0, (5𝑛 + 12)(𝑛 − 3) = 0. This means that 𝑛 − 3 = 0 or 5𝑛 + 12 = 0. The first option gives a solution 𝑛 = 3, the second has no integer solutions. This way, 𝑛 = 𝟑, 𝑚 = 5𝑛 − 3 = 𝟏𝟐. Denote the width as 𝑤, then the length 𝑙 is eq ...
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