The parabola y = x^2 + 4x + 5 and the line y = mx have no points of intersection...

1. TRANSLATE the problem information

• We need to find when \(\mathrm{y = x^2 + 4x + 5}\) and \(\mathrm{y = mx}\) have NO intersection points
• This means we're looking for values of m where these graphs don't cross each other

2. INFER the mathematical approach

• Intersection points occur when the y-values are equal: \(\mathrm{x^2 + 4x + 5 = mx}\)
• For NO intersection points, this equation must have no real solutions
• This becomes a quadratic equation: \(\mathrm{x^2 + (4-m)x + 5 = 0}\)

3. INFER the solution condition

• A quadratic has no real solutions when its discriminant is negative
• For \(\mathrm{ax^2 + bx + c = 0}\), discriminant = \(\mathrm{b^2 - 4ac}\)
• Here: \(\mathrm{a = 1, b = (4-m), c = 5}\)
• So discriminant = \(\mathrm{(4-m)^2 - 4(1)(5) = (4-m)^2 - 20}\)

4. SIMPLIFY the discriminant inequality

• For no real solutions: \(\mathrm{(4-m)^2 - 20 \lt 0}\)
• Rearranging: \(\mathrm{(4-m)^2 \lt 20}\)
• Taking square root: \(\mathrm{|4-m| \lt \sqrt{20} = 2\sqrt{5} \approx 4.47}\) (use calculator)

5. SIMPLIFY the absolute value inequality

• \(\mathrm{|4-m| \lt 4.47}\) means \(\mathrm{-4.47 \lt 4-m \lt 4.47}\)
• Subtracting 4 from all parts: \(\mathrm{-8.47 \lt -m \lt 0.47}\)
• Multiplying by -1 (and flipping inequalities): \(\mathrm{-0.47 \lt m \lt 8.47}\)

6. APPLY CONSTRAINTS to select the answer

• Among the choices, only \(\mathrm{m = 0}\) falls within the range \(\mathrm{(-0.47, 8.47)}\)
• All other choices (-3, 9, 10) fall outside this range

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often misinterpret "no points of intersection" and incorrectly set up the discriminant condition as ≥ 0 instead of < 0.

They think "no solutions" means the discriminant should be positive, leading them to solve \(\mathrm{(4-m)^2 \geq 20}\) instead. This gives them \(\mathrm{m \leq -0.47}\) or \(\mathrm{m \geq 8.47}\), making them select Choice A (-3) since it satisfies \(\mathrm{m \leq -0.47}\).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that the discriminant should be negative but make algebraic errors when solving \(\mathrm{|4-m| \lt 2\sqrt{5}}\).

Common mistake is forgetting to flip inequality signs when multiplying by -1, or incorrectly handling the absolute value. This leads to confusion about which values of m are valid, causing them to get stuck and guess.

The Bottom Line:

This problem requires careful translation of the geometric condition (no intersection) into an algebraic constraint (negative discriminant), followed by precise inequality manipulation. Students who struggle with either the conceptual connection or the algebraic technique will likely select incorrect answers.