If [tex]x^2+m x+m[/tex] is a perfect-square trinomial, which equation must be true?
[tex]x^2+m x+m=(x+1)^2[/tex]
[tex]x^2+m x+m=(x-1)^2[/tex]
[tex]x^2+m x+m=(x+2)^2[/tex]
[tex]x^2+m x+m=(x+4)^2[/tex]
Answer (1)
Recognize that a perfect square trinomial can be written as ( x + a ) 2 .
Expand ( x + a ) 2 and equate coefficients with x 2 + m x + m to get m = 2 a and m = a 2 .
Solve for a from 2 a = a 2 , which gives a = 0 or a = 2 .
If a = 2 , then m = 4 , and the equation becomes x 2 + 4 x + 4 = ( x + 2 ) 2 . Thus, the correct equation is x 2 + m x + m = ( x + 2 ) 2 .
Explanation
Understanding the Problem We are given that x 2 + m x + m is a perfect-square trinomial. This means it can be written in the form ( x + a ) 2 for some constant a . Our goal is to find which of the given equations must be true.
Expanding the Perfect Square Expanding ( x + a ) 2 , we get x 2 + 2 a x + a 2 . Since x 2 + m x + m is a perfect square trinomial, we can equate it to x 2 + 2 a x + a 2 .
Equating Coefficients Equating the coefficients, we have m = 2 a and m = a 2 . Therefore, 2 a = a 2 , which implies a 2 − 2 a = 0 .
Solving for a Factoring, we get a ( a − 2 ) = 0 , so a = 0 or a = 2 . If a = 0 , then m = 2 a = 0 and m = a 2 = 0 . In this case, x 2 + m x + m = x 2 , which is a perfect square. However, this case is not represented in the options.
Finding the Value of m If a = 2 , then m = 2 a = 4 and m = a 2 = 4 . In this case, x 2 + m x + m = x 2 + 4 x + 4 = ( x + 2 ) 2 . Therefore, the equation x 2 + m x + m = ( x + 2 ) 2 must be true.
Conclusion Thus, the correct equation is x 2 + m x + m = ( x + 2 ) 2 .
Examples
Perfect square trinomials are useful in various applications, such as completing the square to solve quadratic equations or in simplifying algebraic expressions. For example, if you are designing a square garden and need to determine the side length based on the area, recognizing a perfect square trinomial can help you find the dimensions easily. Suppose the area of the garden is represented by x 2 + 6 x + 9 . By recognizing this as ( x + 3 ) 2 , you know that the side length of the square garden is x + 3 .
