To express the function f(x) = x² + 12x + 6 in vertex form, we need to complete the square. The vertex form of a quadratic function is given by:
f(x) = a(x – h)² + k
where (h, k) is the vertex of the parabola.
Starting with the function:
f(x) = x² + 12x + 6
We focus on the x² and 12x terms. To complete the square for these, we take the coefficient of x, which is 12, divide it by 2 to get 6, and then square it to get 36.
Next, we can rewrite the function by adding and subtracting 36:
f(x) = (x² + 12x + 36) – 36 + 6
Which simplifies to:
f(x) = (x + 6)² – 30
Now, we see that the function is in vertex form, f(x) = (x + 6)² – 30, where the vertex is (-6, -30).
Therefore, the zero pair that could be added is +36 and -36, which allows us to balance the equation while completing the square.