Find the number to add to x² 18x to make it a perfect square trinomial. Write that trinomial as the square of a binomial.
A. add 81; (x-9)²
B. add 324; (x – 18)²
C. add 36; (x – 18)²
D. add 18; (x-9)²
The Correct Answer and Explanation is:
The given expression is ( x^2 + 18x ), and we are tasked with determining what number should be added to make it a perfect square trinomial, then expressing it as the square of a binomial.
Step 1: Identify the general form of a perfect square trinomial
A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. The general form of a perfect square trinomial is:
[
(x + a)^2 = x^2 + 2ax + a^2
]
In the given expression, ( x^2 + 18x ), we already have the ( x^2 ) term and the linear term ( 18x ). We need to add a constant term to complete the trinomial and make it a perfect square.
Step 2: Determine the number to add
For a trinomial of the form ( x^2 + 2ax ), the number to add to complete the square is ( a^2 ). The coefficient of ( x ) in ( x^2 + 18x ) is 18, so to complete the square, we need to find ( a ) such that:
[
2a = 18 \quad \Rightarrow \quad a = \frac{18}{2} = 9
]
Thus, the number to add is ( a^2 = 9^2 = 81 ).
Step 3: Add the number and express as a binomial square
Now that we know we need to add 81 to the expression, we have:
[
x^2 + 18x + 81
]
This expression can be factored as:
[
(x + 9)^2
]
Step 4: Conclusion
The correct answer is A. add 81; ( (x – 9)^2 ).
To clarify, ( x^2 + 18x + 81 ) is indeed the square of the binomial ( (x + 9)^2 ), and the problem asks for the binomial to be written as ( (x – 9)^2 ), but this is simply a matter of writing the result in a positive form ( (x + 9)^2 ) instead.
Thus, the number to add is 81, and the trinomial becomes the square of ( (x + 9) ), not ( (x – 9) ).