Theory of Intelligence • First-Principles AI & STEM Education

Step-by-Step Solution

Step 1: Analyze the Leading Coefficient

In the standard quadratic form $f(x) = ax^2 + bx + c$ :

a = 1, \quad b = 6, \quad c = 13

Because the leading coefficient $a = 1 > 0$ , the parabola opens upwards ( $\cup$ ). Therefore, the function has an absolute minimum value at its vertex.

Step 2: Complete the Square

Group the $x$ -terms and add/subtract the square of half the linear coefficient $\left(\frac{b}{2}\right)^2 = \left(\frac{6}{2}\right)^2 = 3^2 = 9$ :

\begin{aligned} f(x) &= (x^2 + 6x) + 13 \\ &= (x^2 + 6x + 9) + 13 - 9 \\ &= (x + 3)^2 + 4 \end{aligned}

Step 3: Determine the Extremum Point

For all real numbers $x \in \mathbb{R}$ , a real square is always non-negative:

(x + 3)^2 \ge 0

Equality $(x + 3)^2 = 0$ is attained if and only if:

x + 3 = 0 \implies x = -3

Step 4: Calculate the Minimum Value

Substituting $x = -3$ :

f(-3) = (-3 + 3)^2 + 4 = 0 + 4 = 4