A perfect square trinomial can be represented by a square model with equivalent length and width. Which polynomial can be represented by a perfect square model?
[tex]x^2-6 x+9[/tex]
[tex]x^2-2 x+4[/tex]
[tex]x^2+5 x+10[/tex]
[tex]x^2+4 x+16[/tex]
Answer (2)
The polynomial that represents a perfect square trinomial is x 2 − 6 x + 9 , as it can be rewritten as ( x − 3 ) 2 . The other polynomials do not fit the criteria for perfect square trinomials. Thus, the correct choice is x 2 − 6 x + 9 .
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Check if x 2 − 6 x + 9 is a perfect square trinomial: ( x − 3 ) 2 = x 2 − 6 x + 9 .
Check if x 2 − 2 x + 4 is a perfect square trinomial: ( x − 1 ) 2 = x 2 − 2 x + 1 e q x 2 − 2 x + 4 .
Check if x 2 + 5 x + 10 is a perfect square trinomial: ( x + 2 5 ) 2 = x 2 + 5 x + 4 25 e q x 2 + 5 x + 10 .
Check if x 2 + 4 x + 16 is a perfect square trinomial: ( x + 2 ) 2 = x 2 + 4 x + 4 e q x 2 + 4 x + 16 .
The only perfect square trinomial is x 2 − 6 x + 9 .
Explanation
Problem Analysis We are given four polynomials and need to determine which one is a perfect square trinomial. A perfect square trinomial can be written in the form ( a x + b ) 2 or ( a x − b ) 2 . Let's examine each option.
Checking Each Polynomial
x 2 − 6 x + 9 : We want to see if this can be written as ( x − b ) 2 . If we expand ( x − b ) 2 , we get x 2 − 2 b x + b 2 . Comparing this to x 2 − 6 x + 9 , we see that − 2 b = − 6 , so b = 3 . Then b 2 = 3 2 = 9 , which matches the constant term. Thus, x 2 − 6 x + 9 = ( x − 3 ) 2 , which is a perfect square trinomial.
x 2 − 2 x + 4 : We want to see if this can be written as ( x − b ) 2 . If we expand ( x − b ) 2 , we get x 2 − 2 b x + b 2 . Comparing this to x 2 − 2 x + 4 , we see that − 2 b = − 2 , so b = 1 . Then b 2 = 1 2 = 1 , but the constant term is 4, so this is not a perfect square trinomial.
x 2 + 5 x + 10 : We want to see if this can be written as ( x + b ) 2 . If we expand ( x + b ) 2 , we get x 2 + 2 b x + b 2 . Comparing this to x 2 + 5 x + 10 , we see that 2 b = 5 , so b = 2 5 . Then b 2 = ( 2 5 ) 2 = 4 25 = 6.25 , but the constant term is 10, so this is not a perfect square trinomial.
x 2 + 4 x + 16 : We want to see if this can be written as ( x + b ) 2 . If we expand ( x + b ) 2 , we get x 2 + 2 b x + b 2 . Comparing this to x 2 + 4 x + 16 , we see that 2 b = 4 , so b = 2 . Then b 2 = 2 2 = 4 , but the constant term is 16, so this is not a perfect square trinomial.
Conclusion Therefore, the only perfect square trinomial among the given options is x 2 − 6 x + 9 .
Examples
Perfect square trinomials are useful in various applications, such as completing the square to solve quadratic equations or simplifying algebraic expressions. For example, consider a square garden whose area is represented by the expression x 2 + 8 x + 16 . Recognizing this as a perfect square trinomial, ( x + 4 ) 2 , allows you to determine that the side length of the garden is x + 4 . This concept is also used in engineering to design structures with specific dimensions and properties.
