The graph of the function \(\mathrm{f(x) = x^2 + 8x + k}\) intersects the horizontal line y = 3 at...

Part 1: Brief Solution

Concepts tested: Quadratic discriminant analysis and real solutions

Primary process skills: Infer, Apply Constraints

Key Steps:

• Set up discriminant: For \(64\mathrm{x}^2 + \mathrm{bx} + 25 = 0\), discriminant = \(\mathrm{b}^2 - 4(64)(25) = \mathrm{b}^2 - 6400\)
• Apply constraint for 'more than one real solution': \(\mathrm{b}^2 - 6400 \gt 0\) (not \(\geq 0\))
• Solve inequality: \(\mathrm{b}^2 \gt 6400 \rightarrow |\mathrm{b}| \gt 80 \rightarrow \mathrm{b} \gt 80\) or \(\mathrm{b} \lt -80\)
• Check options: Only A) -91 satisfies \(\mathrm{b} \lt -80\)

Answer: A) -91

Part 2: Top 3 Faltering Points

Top 3 Faltering Points:

1. Misinterpreting 'More Than One' - Phase: Devising Approach → Choice B (-80)
   • Process skill failure: Apply Constraints
   • Students use \(\mathrm{b}^2 - 6400 \geq 0\) instead of \(\gt 0\), incorrectly including the boundary case of exactly one solution.
2. Square Root Calculation Error - Phase: Executing Approach → Choice B (-80) or Various
   • Computational error
   • Students incorrectly calculate \(\sqrt{6400}\) as 64 instead of 80, leading to wrong boundary values.
3. Incomplete Inequality Solution - Phase: Executing Approach → Choice D (40)
   • Process skill failure: Consider All Cases
   • Students solve \(|\mathrm{b}| \gt 80\) as only \(\mathrm{b} \gt 80\), forgetting the negative case \(\mathrm{b} \lt -80\).

Part 3: Detailed Solution

To solve this problem, I need to determine when a quadratic equation has more than one real solution.

Process Skill: TRANSLATE - Converting the English phrase 'more than one real solution' into mathematical terms means I need exactly two distinct real solutions, which occurs when the discriminant is positive (\(\gt 0\)).

For any quadratic equation \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c} = 0\), the discriminant is \(\mathrm{b}^2 - 4\mathrm{ac}\). The number of real solutions depends on this discriminant:

• If discriminant \(\gt 0\): two distinct real solutions
• If discriminant = 0: exactly one real solution (repeated root)
• If discriminant \(\lt 0\): no real solutions

Process Skill: INFER - Since we want 'more than one real solution,' we need the discriminant to be strictly greater than zero, not just non-negative.

In our equation \(64\mathrm{x}^2 + \mathrm{bx} + 25 = 0\):

• a = 64
• coefficient of x = b (the variable we're solving for)
• c = 25

Setting up the discriminant:

Discriminant = \(\mathrm{b}^2 - 4\mathrm{ac}\)

\(= \mathrm{b}^2 - 4(64)(25)\)

\(= \mathrm{b}^2 - 6400\)

Process Skill: APPLY CONSTRAINTS - The phrase 'more than one real solution' establishes the constraint that our discriminant must be strictly positive, not just non-negative.

For more than one real solution:

\(\mathrm{b}^2 - 6400 \gt 0\)

\(\mathrm{b}^2 \gt 6400\)

Taking the square root of both sides:

\(|\mathrm{b}| \gt \sqrt{6400}\)

Process Skill: SIMPLIFY - Let me calculate \(\sqrt{6400}\):

\(\sqrt{6400} = \sqrt{64 \times 100}\)

\(= \sqrt{64} \times \sqrt{100}\)

\(= 8 \times 10\)

\(= 80\)

Therefore: \(|\mathrm{b}| \gt 80\)

Process Skill: CONSIDER ALL CASES - The absolute value inequality \(|\mathrm{b}| \gt 80\) means:

\(\mathrm{b} \gt 80\) OR \(\mathrm{b} \lt -80\)

Now I'll check each option:

• A) b = -91: Since -91 < -80, this satisfies our condition ✓
• B) b = -80: Since -80 is the boundary (not < -80), this gives exactly one solution ✗
• C) b = 5: Since -80 < 5 < 80, this gives no real solutions ✗
• D) b = 40: Since -80 < 40 < 80, this gives no real solutions ✗

Verification for option A:

When b = -91: Discriminant = \((-91)^2 - 6400\)

\(= 8281 - 6400\)

\(= 1881 \gt 0\) ✓

The answer is A) -91.

Part 4: Detailed Faltering Points Analysis

Errors while devising the approach:

• Misinterpreting 'More Than One': Students may think 'more than one real solution' means 'at least one real solution,' leading them to use discriminant \(\geq 0\) instead of \(\gt 0\). This process skill failure in Apply Constraints would incorrectly include option B (-80), which gives exactly one solution.
• Confusing Discriminant Formula: Some students might incorrectly identify the coefficients, using the wrong values for a, b, or c in the discriminant formula. This Translate failure leads to completely wrong calculations.

Errors while executing the approach:

• Square Root Calculation Error: Students might incorrectly calculate \(\sqrt{6400}\) as 64 instead of 80, possibly by confusing \(\sqrt{64} = 8\) with the full calculation. This computational error leads to wrong boundary values (\(|\mathrm{b}| \gt 64\) instead of \(|\mathrm{b}| \gt 80\)).
• Incomplete Case Analysis: When solving \(|\mathrm{b}| \gt 80\), students might only consider \(\mathrm{b} \gt 80\) and forget about \(\mathrm{b} \lt -80\). This Consider All Cases failure would eliminate the correct answer A (-91) from consideration.
• Inequality Direction Error: Students might flip inequality signs incorrectly when moving from \(\mathrm{b}^2 \gt 6400\) to \(|\mathrm{b}| \gt 80\), leading to wrong solution sets.

Errors while selecting the answer:

• Boundary Confusion: Students who correctly find \(|\mathrm{b}| \gt 80\) might still select option B (-80) by incorrectly thinking the boundary case counts as 'more than one solution.' This reflects a failure to Apply Constraints properly.
• Sign Errors: Students might correctly identify that \(|\mathrm{b}| \gt 80\), but then select option D (40) by focusing only on positive values and missing that negative values below -80 also work.