\(\mathrm{f(x) = 4x^2 - 50x + 126}\) The given equation defines the function f. For what value of x does...

1. INFER what the problem is asking

• Given: \(\mathrm{f(x) = 4x^2 - 50x + 126}\)
• Find: The x-value where \(\mathrm{f(x)}\) reaches its minimum
• Key insight: This is a quadratic function, and we need to find its vertex

2. INFER the solution approach

• Since the coefficient of \(\mathrm{x^2}\) is positive (\(\mathrm{a = 4 \gt 0}\)), the parabola opens upward
• This means the function has a minimum value at its vertex
• We need to find the x-coordinate of the vertex

3. SIMPLIFY using the vertex formula

• For \(\mathrm{f(x) = ax^2 + bx + c}\), the vertex occurs at \(\mathrm{x = -b/(2a)}\)
• Here: \(\mathrm{a = 4, b = -50, c = 126}\)
• Calculate: \(\mathrm{x = -(-50)/(2×4) = 50/8 = 25/4}\)

4. SIMPLIFY to final form

• \(\mathrm{25/4 = 6.25}\) (use calculator if needed)
• Both forms are acceptable answers

Answer: 25/4 or 6.25

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that "minimum of a quadratic" means "find the vertex"

Many students see the quadratic and think they need to solve \(\mathrm{f(x) = 0}\) or set up some equation. They may try to factor the quadratic or use the quadratic formula, not realizing this problem is about optimization, not solving. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Sign errors when applying the vertex formula \(\mathrm{x = -b/(2a)}\)

Students correctly identify they need the vertex formula but make arithmetic mistakes. Common errors include:

• Forgetting the negative sign: using \(\mathrm{x = b/(2a)}\) instead of \(\mathrm{x = -b/(2a)}\)
• Getting confused with \(\mathrm{-(-50)}\) and writing \(\mathrm{x = -50/(2×4) = -6.25}\)

This may lead them to select an incorrect answer or become confused about negative values.

The Bottom Line:

This problem tests whether students can connect "finding a minimum" with "finding a vertex." The calculation itself is straightforward once they make this conceptual leap, but many students get stuck because they don't recognize this as an optimization problem rather than an equation-solving problem.