Final answer:
The value of k that makes the expression kx^2 - 198x + 81 a perfect square trinomial is 11, provided by setting 2ab = -198 and b^2 = 81, solving for k and b simultaneously.
Step-by-step explanation:
To determine the possible values of k that would make the expression kx^2 - 198x + 81 a perfect square trinomial, we need to find a value for k so that the x-term can be expressed as the square of half the coefficient of the x-term.
A perfect square trinomial takes the form (ax+b)^2 = a^2x^2 + 2abx + b^2. To make the given expression a perfect square trinomial, 2ab must equal -198 and b^2 must equal 81. By taking the square root of 81, we get b = ±9. Since we need a negative middle term (because our middle term is -198), we'll use -9 for b.
k would then be computed using the relationship 2ab = -198, giving us k = -198/(2*(-9)) which simplifies to k = -198/-18 = 11. As such, k must be 11 for the expression to be a perfect square trinomial.