Which value must be added to the expression [tex]x^2+16 x[/tex] to make it a perfect-square trinomial?
A. 8
B. 32
C. 64
D. 256
Answer (2)
We have the expression x 2 + 16 x .
We want to find a value to add to make it a perfect square trinomial of the form ( x + a ) 2 = x 2 + 2 a x + a 2 .
We find a by setting 2 a = 16 , which gives a = 8 .
We calculate a 2 = 8 2 = 64 , which is the value to be added. The final answer is 64 .
Explanation
Understanding the Problem We are given the expression x 2 + 16 x . We want to find a value to add to this expression 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 + 16 x to the form x 2 + 2 a x . We want to find the value of a such that 2 a = 16 .
Calculating a Dividing both sides of the equation 2 a = 16 by 2, we get a = 2 16 = 8 .
Calculating a^2 Now we calculate a 2 , which is the value that needs to be added to complete the perfect-square trinomial. a 2 = 8 2 = 64 .
Conclusion Thus, the perfect square trinomial is x 2 + 16 x + 64 = ( x + 8 ) 2 . Therefore, the value that must be added to the expression x 2 + 16 x to make it a perfect-square trinomial is 64.
Examples
Perfect square trinomials are useful in many areas of mathematics, such as solving quadratic equations, completing the square, and simplifying expressions. For example, if you are building a square garden and you know one side must be 'x' meters long, and you have an extra 16 meters of fencing, you can use perfect square trinomials to determine how much additional area you need to make the garden a perfect square.
To make the expression x 2 + 16 x a perfect-square trinomial, we must add 64. This is found by determining that a is 8 from the formula 2 a = 16 and then calculating a 2 = 64 . Therefore, the correct answer is 64.
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