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)
We want to find a value to add to x 2 + 12 x to make it a perfect-square trinomial.
We compare the given expression to the form x 2 + 2 a x + a 2 and set 2 a = 12 .
We solve for a and find a = 6 .
We calculate a 2 = 36 , which is the value that must be added to the expression. The final answer 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 has the form ( x + a ) 2 = x 2 + 2 a x + a 2 .
Finding the Value of a We compare the given expression x 2 + 12 x to the form x 2 + 2 a x + a 2 . We want to find the value of a such that 2 a = 12 .
Solving for a We solve for a in the equation 2 a = 12 . Dividing both sides by 2, we get a = 2 12 = 6 .
Calculating a^2 Now we calculate a 2 , which is the value that needs to be added to the expression to make it a perfect-square trinomial. We have a 2 = 6 2 = 36 . Therefore, the value that must be added to the expression x 2 + 12 x to make it a perfect-square trinomial is 36.
Verification To verify, we can check that x 2 + 12 x + 36 is indeed a perfect-square trinomial. We have x 2 + 12 x + 36 = ( x + 6 ) 2 , which is a perfect square.
Examples
Perfect square trinomials are useful in many areas of mathematics, such as solving quadratic equations, completing the square, and simplifying expressions. For example, consider a rectangular garden whose area is given by the expression x 2 + 12 x . By adding 36 to this expression, we can rewrite the area as ( x + 6 ) 2 , which represents a square garden with side length x + 6 . This can be useful in determining the dimensions of the garden or in calculating the amount of fencing needed to enclose it.
To make the expression x 2 + 12 x a perfect-square trinomial, we need to add 36. This is determined by solving for the value of a in the perfect-square formula. The final answer is option B: 36.
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