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 (2)
Recognize that a perfect square trinomial can be written as ( x + a ) 2 .
Expand ( x + a ) 2 and equate coefficients to find m = 2 a and m = a 2 .
Solve for a to find a = 0 or a = 2 , which gives m = 0 or m = 4 .
Check the given options to find the equation that holds true, which is x 2 + m x + m = ( x + 2 ) 2 .
The equation that must be true is x 2 + m x + m = ( x + 2 ) 2 .
Explanation
Understanding the Problem We are given a quadratic expression x 2 + m x + m and told that it is a perfect-square trinomial. Our goal is to determine which of the given equations must be true. A perfect square trinomial can be written in the form ( x + a ) 2 = x 2 + 2 a x + a 2 . Therefore, if x 2 + m x + m is a perfect square trinomial, then x 2 + m x + m = ( x + a ) 2 for some a .
Finding Possible Values of a Expanding ( x + a ) 2 gives x 2 + 2 a x + a 2 . Equating the coefficients of x 2 + m x + m and x 2 + 2 a x + a 2 gives m = 2 a and m = a 2 . Therefore, 2 a = a 2 , which implies a 2 − 2 a = 0 , so a ( a − 2 ) = 0 . Thus, a = 0 or a = 2 .
Determining Possible Forms of the Trinomial If a = 0 , then m = 2 a = 0 and m = a 2 = 0 . So x 2 + m x + m = x 2 , which is a perfect square. If a = 2 , then m = 2 a = 4 and m = a 2 = 4 . So x 2 + m x + m = x 2 + 4 x + 4 = ( x + 2 ) 2 . Therefore, x 2 + m x + m = ( x + 2 ) 2 must be true when m = 4 .
Checking the Options Now, let's check the given options to see which equation must be true.
Option 1: x 2 + m x + m = ( x − 1 ) 2 = x 2 − 2 x + 1 . Then m = − 2 and m = 1 , which is impossible. Option 2: x 2 + m x + m = ( x + 1 ) 2 = x 2 + 2 x + 1 . Then m = 2 and m = 1 , which is impossible. Option 3: x 2 + m x + m = ( x + 2 ) 2 = x 2 + 4 x + 4 . Then m = 4 and m = 4 , which is possible. Option 4: x 2 + m x + m = ( x + 4 ) 2 = x 2 + 8 x + 16 . Then m = 8 and m = 16 , which is impossible.
Therefore, the equation x 2 + m x + m = ( x + 2 ) 2 must be true.
Examples
Perfect square trinomials are useful in many areas of mathematics, such as completing the square to solve quadratic equations, simplifying algebraic expressions, and solving problems in geometry. For example, if you are designing a square garden and need to determine the side length based on the area, understanding perfect square trinomials can help you find the dimensions efficiently. They also appear in physics, such as in the study of projectile motion, where the height of an object can be modeled by a quadratic equation.
The equation that must be true for the expression x 2 + m x + m to be a perfect-square trinomial is x 2 + m x + m = ( x + 2 ) 2 . This holds when m = 4 .
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