The area of a square is $9 x^2+12 x+4$.
a) Determine an expression that represents the perimeter. Note: Do not use the quadratic formula.
b) State an equivalent expression to the perimeter in question 3(a).
Answer (2)
Factor the area expression: 9 x 2 + 12 x + 4 = ( 3 x + 2 ) 2 .
Determine the side length: ( 3 x + 2 ) 2 = 3 x + 2 .
Calculate the perimeter: 4 ( 3 x + 2 ) .
Simplify the perimeter expression: 12 x + 8 . The perimeter is 12 x + 8 .
Explanation
Understanding the Problem The area of a square is given as 9 x 2 + 12 x + 4 . We need to find an expression for the perimeter of the square. The perimeter is the sum of all the sides of the square. Since all sides of a square are equal, the perimeter is four times the length of one side. To find the side length, we need to take the square root of the area.
Finding the Side Length The area of the square is 9 x 2 + 12 x + 4 . We can factor this expression to find the side length. Notice that 9 x 2 + 12 x + 4 is a perfect square trinomial. It can be written as ( 3 x ) 2 + 2 ( 3 x ) ( 2 ) + ( 2 ) 2 . This factors to ( 3 x + 2 ) 2 .
Calculating the Square Root Since the area of the square is ( 3 x + 2 ) 2 , the side length of the square is the square root of the area, which is ( 3 x + 2 ) 2 = 3 x + 2 .
Calculating the Perimeter Now that we have the side length, we can find the perimeter. The perimeter of a square is 4 times the side length. So, the perimeter is 4 ( 3 x + 2 ) .
Simplifying the Expression To find an equivalent expression for the perimeter, we distribute the 4: 4 ( 3 x + 2 ) = 4 ( 3 x ) + 4 ( 2 ) = 12 x + 8 . Therefore, the expression that represents the perimeter is 12 x + 8 .
Final Answer The expression that represents the perimeter of the square is 12 x + 8 .
Examples
Imagine you're designing a square garden and you know the area can be represented by the expression 9 x 2 + 12 x + 4 . To build a fence around the garden, you need to know the perimeter. By finding the square root of the area, you determine the length of one side. Then, multiplying that side length by 4 gives you the total length of fencing material needed. This is a practical application of factoring and finding perimeters in real-world construction and design scenarios.
The perimeter of the square is given by the expression 12 x + 8 , found by factoring the area and calculating the side length. The side length of the square is 3 x + 2 . Therefore, the final perimeter expression is 12 x + 8 .
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