What value of k makes the expression P^2 - 18P + K a perfect square?
81
To determine the value of k that makes the expression P^2 - 18P + k a perfect square, we can follow these steps:
Step 1: Recognize the perfect square trinomial form.
A perfect square trinomial is expressed in the format (a + b)^2, where a and b are constants. Therefore, if we can rewrite the given expression as (P - a)^2, we can determine the value of k.
Step 2: Expand the perfect square form.
Expanding (P - a)^2 will give us:
(P - a)(P - a) = P^2 - 2aP + a^2
Step 3: Compare coefficients.
Comparing the expanded form P^2 - 2aP + a^2 with the given expression P^2 - 18P + k, we can equate the coefficients of the P terms:
-2aP = -18P --> 2a = 18 --> a = 9
Step 4: Determine the value of k.
Now that we have found the value of a, we can substitute it back into the expanded form and equate it to the given expression to find k:
P^2 - 2aP + a^2 = P^2 - 18P + k
P^2 - 2(9)P + 9^2 = P^2 - 18P + k
P^2 - 18P + 81 = P^2 - 18P + k
Since the P^2 and -18P terms are the same on both sides, we can conclude that k = 81.
Therefore, the value of k that makes the expression P^2 - 18P + k a perfect square is k = 81.