Study the products shown. Is there a pattern?
[tex]\begin{array}{l}
(x+3)^2=x^2+6 x+9 \\
(x+4)^2=x^2+8 x+16 \\
(x+5)^2=x^2+10 x+25 \\
(x+6)^2=x^2+12 x+36
\end{array}[/tex]
Each product is in the form [tex]$a x^2+b x+c$[/tex]. Which of the following describes the relationship between [tex]$b$[/tex] and [tex]$c$[/tex]?
A. [tex]$c$[/tex] is the square of half of [tex]$b$[/tex].
B. [tex]$c$[/tex] is the square of [tex]$b$[/tex]
C. [tex]$c$[/tex] is 1.5 times that of [tex]$b$[/tex].
Answer (2)
The relationship between b and c in the given quadratic expansions is that c is the square of half of b . This can be verified by checking each expansion:
( x + 3 ) 2 = x 2 + 6 x + 9 : ( 6/2 ) 2 = 9
( x + 4 ) 2 = x 2 + 8 x + 16 : ( 8/2 ) 2 = 16
( x + 5 ) 2 = x 2 + 10 x + 25 : ( 10/2 ) 2 = 25
( x + 6 ) 2 = x 2 + 12 x + 36 : ( 12/2 ) 2 = 36
Thus, the correct answer is:
c is the square of half of b
Explanation
Understanding the Problem We are given the following expansions:
( x + 3 ) 2 = x 2 + 6 x + 9 ( x + 4 ) 2 = x 2 + 8 x + 16 ( x + 5 ) 2 = x 2 + 10 x + 25 ( x + 6 ) 2 = x 2 + 12 x + 36
Each product is in the form a x 2 + b x + c . We need to find the relationship between b and c . The possible relationships are:
c is the square of half of b .
c is the square of b
c is 1.5 times that of b .
Analyzing the First Expansion Let's analyze the first expansion ( x + 3 ) 2 = x 2 + 6 x + 9 . Here, b = 6 and c = 9 .
Is c the square of half of b ? ( b /2 ) 2 = ( 6/2 ) 2 = 3 2 = 9 . Since c = 9 , this relationship holds.
Is c the square of b ? b 2 = 6 2 = 36 . Since c = 9 , this relationship does not hold.
Is c 1.5 times that of b ? 1.5 b = 1.5 ( 6 ) = 9 . Since c = 9 , this relationship holds.
So, for the first expansion, both relationship 1 and 3 hold.
Analyzing the Second Expansion Now let's analyze the second expansion ( x + 4 ) 2 = x 2 + 8 x + 16 . Here, b = 8 and c = 16 .
Is c the square of half of b ? ( b /2 ) 2 = ( 8/2 ) 2 = 4 2 = 16 . Since c = 16 , this relationship holds.
Is c the square of b ? b 2 = 8 2 = 64 . Since c = 16 , this relationship does not hold.
Is c 1.5 times that of b ? 1.5 b = 1.5 ( 8 ) = 12 . Since c = 16 , this relationship does not hold.
So, for the second expansion, only relationship 1 holds.
Analyzing the Third Expansion Now let's analyze the third expansion ( x + 5 ) 2 = x 2 + 10 x + 25 . Here, b = 10 and c = 25 .
Is c the square of half of b ? ( b /2 ) 2 = ( 10/2 ) 2 = 5 2 = 25 . Since c = 25 , this relationship holds.
Is c the square of b ? b 2 = 1 0 2 = 100 . Since c = 25 , this relationship does not hold.
Is c 1.5 times that of b ? 1.5 b = 1.5 ( 10 ) = 15 . Since c = 25 , this relationship does not hold.
So, for the third expansion, only relationship 1 holds.
Analyzing the Fourth Expansion Now let's analyze the fourth expansion ( x + 6 ) 2 = x 2 + 12 x + 36 . Here, b = 12 and c = 36 .
Is c the square of half of b ? ( b /2 ) 2 = ( 12/2 ) 2 = 6 2 = 36 . Since c = 36 , this relationship holds.
Is c the square of b ? b 2 = 1 2 2 = 144 . Since c = 36 , this relationship does not hold.
Is c 1.5 times that of b ? 1.5 b = 1.5 ( 12 ) = 18 . Since c = 36 , this relationship does not hold.
So, for the fourth expansion, only relationship 1 holds.
Conclusion From the above analysis, we can conclude that the relationship between b and c is that c is the square of half of b .
Examples
Completing the square is a useful technique in algebra. For example, consider a farmer who wants to build a rectangular enclosure for his animals. He has a fixed amount of fencing, and he wants to maximize the area of the enclosure. By expressing the area in terms of the length of one side and then completing the square, the farmer can find the dimensions that maximize the area. This is a practical application of completing the square in optimization problems.
The relationship between the coefficients b and c in the quadratic expansions is that c is the square of half of b . This holds true for all the provided expansions. Therefore, the answer is option A.
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