A quadratic function is a perfect square binomial if it can be expressed in the form \((x + a)^2\), where \(a\) is a constant. This corresponds to the expanded form \(x^2 + 2ax + a^2\). To determine which of the given functions is a perfect square binomial, we can check if the first two terms can form this structure.
Let's analyze each option:
A. \(f(x) = x^2 + 6x + 8\)
- The first two terms are \(x^2 + 6x\). Here, \(2a = 6 \Rightarrow a = 3\), and \(a^2 = 9\). The constant term is not equal to \(a^2\), so this is not a perfect square.
B. \(f(x) = x^2 + 7x + 12\)
- The first two terms are \(x^2 + 7x\). Here, \(2a = 7 \Rightarrow a = 3.5\), and \(a^2 = 12.25\). The constant term is not equal to \(a^2\), so this is not a perfect square.
C. \(f(x) = x^2 - 9x + 18\)
- The first two terms are \(x^2 - 9x\). Here, \(2a = -9 \Rightarrow a = -4.5\), and \(a^2 = 20.25\). The constant term is not equal to \(a^2\), so this is not a perfect square.
D. \(f(x) = x^2 + 10x + 25\)
- The first two terms are \(x^2 + 10x\). Here, \(2a = 10 \Rightarrow a = 5\), and \(a^2 = 25\). The constant term is equal to \(a^2\), so this factors to \((x + 5)^2\).
The correct answer is: D. \(f(x) = x^2 + 10x + 25\), which is a perfect square binomial.