For real numbers a, the expression \(\mathrm{f(a) = a^2 - 8a + 73}\) represents a function. What is the minimum...

1. INFER the problem strategy

• Given: \(\mathrm{f(a) = a^2 - 8a + 73}\), find minimum value
• Since this is a quadratic with positive leading coefficient, it opens upward and has a minimum
• Strategy: Complete the square to find vertex form, which reveals the minimum directly

2. SIMPLIFY by completing the square

• Start with \(\mathrm{f(a) = a^2 - 8a + 73}\)
• Take the coefficient of a: \(\mathrm{-8}\)
• Divide by 2: \(\mathrm{-8 ÷ 2 = -4}\)
• Square the result: \(\mathrm{(-4)^2 = 16}\)
• Add and subtract 16: \(\mathrm{f(a) = a^2 - 8a + 16 + 73 - 16}\)

3. SIMPLIFY to vertex form

• Group the perfect square trinomial: \(\mathrm{f(a) = (a^2 - 8a + 16) + 73 - 16}\)
• Factor the trinomial: \(\mathrm{f(a) = (a - 4)^2 + 57}\)

4. INFER the minimum value location

• In vertex form \(\mathrm{f(a) = (a - 4)^2 + 57}\)
• Since \(\mathrm{(a - 4)^2 \geq 0}\) for all real numbers a, the minimum value of this expression is 0
• The minimum occurs when \(\mathrm{(a - 4)^2 = 0}\), which happens when \(\mathrm{a = 4}\)

5. SIMPLIFY to find the minimum value

• When \(\mathrm{a = 4}\): \(\mathrm{f(4) = (4 - 4)^2 + 57 = 0 + 57 = 57}\)

Answer: B (57)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may try to find the minimum using calculus (taking the derivative) or by plugging in answer choices, rather than recognizing that completing the square is the most direct algebraic approach.

This leads to unnecessary complexity and potential calculation errors, causing them to abandon systematic solution and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when completing the square, particularly when adding and subtracting the same value (16 in this case) or when factoring the perfect square trinomial.

Common mistake: Writing \(\mathrm{(a + 4)^2}\) instead of \(\mathrm{(a - 4)^2}\), leading to \(\mathrm{f(a) = (a + 4)^2 + 89}\). This may lead them to select Choice D (89).

The Bottom Line:

The key insight is recognizing that completing the square transforms a quadratic into a form where the minimum value is immediately visible - you don't need calculus or trial-and-error with answer choices.