\(\mathrm{f(x) = (x - 14)(x + 19)}\)The function f is defined by the given equation. For what value of x...

1. INFER the problem type and strategy

• Given: \(\mathrm{f(x) = (x - 14)(x + 19)}\)
• We need the x-value where \(\mathrm{f(x)}\) reaches its minimum
• Since this is a quadratic function, the minimum occurs at the vertex (assuming the parabola opens upward)

2. INFER that we need the vertex x-coordinate

• For any quadratic function, we can use the vertex formula: \(\mathrm{x = -\frac{b}{2a}}\)
• First, we need to expand to standard form \(\mathrm{ax^2 + bx + c}\)

3. SIMPLIFY by expanding the factored form

• \(\mathrm{f(x) = (x - 14)(x + 19)}\)
• \(\mathrm{f(x) = x^2 + 19x - 14x - 266}\)
• \(\mathrm{f(x) = x^2 + 5x - 266}\)
• Now we have: \(\mathrm{a = 1, b = 5, c = -266}\)

4. SIMPLIFY using the vertex formula

• x-coordinate of vertex = \(\mathrm{-\frac{b}{2a}}\)
• \(\mathrm{x = -\frac{5}{2 \cdot 1} = -\frac{5}{2}}\)

5. Verify the parabola opens upward

• Since \(\mathrm{a = 1 \gt 0}\), the parabola opens upward
• Therefore, the vertex is indeed a minimum point

Answer: D. -5/2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect "minimum value" with "vertex of parabola." They might try to set the function equal to zero and solve for x-intercepts instead, getting \(\mathrm{x = 14}\) or \(\mathrm{x = -19}\). Since neither of these appears in the answer choices, this leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when expanding \(\mathrm{(x - 14)(x + 19)}\). They might get \(\mathrm{x^2 - 5x - 266}\) instead of \(\mathrm{x^2 + 5x - 266}\), leading to \(\mathrm{x = -\frac{(-5)}{2 \cdot 1} = \frac{5}{2}}\). Since \(\mathrm{\frac{5}{2}}\) isn't among the choices, this may lead them to select Choice C \(\mathrm{(-\frac{33}{2})}\) thinking they made a small arithmetic error.

The Bottom Line:

This problem requires recognizing that finding a minimum means finding a vertex, then executing the algebra carefully. The key insight is connecting the language of "minimum" to the mathematical concept of vertex.