Which value must be added to the expression [tex]x^2+12 x[/tex] to make it a perfect-square trinomial?
A. 6
B. 36
C. 72
D. 144
Answer (2)
Recognize the form of a perfect-square trinomial: ( x + a ) 2 = x 2 + 2 a x + a 2 .
Determine the value of a by comparing the given expression to the perfect-square trinomial form: 2 a = 12 , so a = 6 .
Calculate a 2 to find the value to be added: a 2 = 6 2 = 36 .
The value that must be added to x 2 + 12 x to make it a perfect-square trinomial is 36 .
Explanation
Understanding the Problem We are given the expression x 2 + 12 x and we want to find the value that must be added to it to make it a perfect-square trinomial. A perfect-square trinomial is a trinomial that can be factored into the square of a binomial.
Relating to Perfect-Square Trinomial Form A perfect-square trinomial has the form ( x + a ) 2 = x 2 + 2 a x + a 2 . We want to find a value to add to x 2 + 12 x so that it matches this form.
Finding the Value of a Comparing x 2 + 12 x to x 2 + 2 a x , we can see that 2 a = 12 .
Solving for a Solving for a , we divide both sides of the equation 2 a = 12 by 2: a = 2 12 = 6
Calculating a^2 Now we need to find a 2 , which is the value that must be added to complete the perfect-square trinomial. Since a = 6 , we have: a 2 = 6 2 = 36
Conclusion Therefore, the value that must be added to x 2 + 12 x to make it a perfect-square trinomial is 36. The perfect-square trinomial is x 2 + 12 x + 36 , which can be factored as ( x + 6 ) 2 .
Examples
Perfect square trinomials are useful in many areas of mathematics, such as completing the square to solve quadratic equations, simplifying expressions, and solving optimization problems. For example, consider a rectangular garden with one side of length x and the other side of length x + 12 . To find the area needed to make it a perfect square, we would need to add 36 square units, resulting in a square garden with sides of length x + 6 .
The value that must be added to the expression x 2 + 12 x to create a perfect-square trinomial is 36 . This means the expression becomes x 2 + 12 x + 36 = ( x + 6 ) 2 . Therefore, the correct answer is option B: 36.
;
