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)
A perfect square trinomial x 2 + m x + m can be written as ( x + a ) 2 = x 2 + 2 a x + a 2 .
Equating coefficients, we have m = 2 a and m = a 2 .
Setting 2 a = a 2 , we find a = 0 or a = 2 .
If a = 0 , then m = 0 , and x 2 + m x + m = x 2 .
If a = 2 , then m = 4 , and x 2 + m x + m = ( x + 2 ) 2 . Thus, the equation that must be true 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.
Setting up Equations Expanding ( x + a ) 2 , we get x 2 + 2 a x + a 2 . Comparing this with x 2 + m x + m , we have two equations:
m = 2 a (1) m = a 2 (2)
Solving for a From equations (1) and (2), we have 2 a = a 2 . Rearranging, we get a 2 − 2 a = 0 . Factoring out a , we have a ( a − 2 ) = 0 . This gives us two possible values for a : a = 0 or a = 2 .
Finding m If a = 0 , then from equation (1), m = 2 ( 0 ) = 0 . In this case, x 2 + m x + m = x 2 + 0 x + 0 = x 2 , which is indeed a perfect square ( x + 0 ) 2 = x 2 .
If a = 2 , then from equation (1), m = 2 ( 2 ) = 4 . In this case, x 2 + m x + m = x 2 + 4 x + 4 = ( x + 2 ) 2 , which is a perfect square.
Checking the Options Now we check which of the given equations must be true. We found that either m = 0 and x 2 + m x + m = x 2 , or m = 4 and x 2 + m x + m = ( x + 2 ) 2 .
If m = 0 , then x 2 + m x + m = x 2 , which is not equal to ( x − 1 ) 2 , ( x + 1 ) 2 , or ( x + 4 ) 2 . However, if m = 4 , then x 2 + m x + m = x 2 + 4 x + 4 = ( x + 2 ) 2 , which matches the third option.
Final Answer Therefore, the equation that must be true 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, simplifying algebraic expressions, and modeling physical phenomena. For example, in physics, the equation of motion for a projectile can sometimes be expressed as a perfect square trinomial, allowing for easy determination of the time at which the projectile reaches its maximum height. Understanding perfect square trinomials helps in optimizing designs and predicting outcomes in engineering and physics problems.
