SAT Math: Line y = -4x + k Intersects Parabola y = 2x² + 8x - 5 at Exactly One Point
Set 2x^2 + 8x - 5 = -4x + k => 2x^2 + 12x - (5 + k) = 0. Delta = 12^2 - 4(2)(-5 - k) = 0 => 144 + 40 + 8k = 0 => 184 + 8k = 0 => k = -23.
Exact Question Text
In the xy-plane, the line y = -4x + k intersects the parabola y = 2x^2 + 8x - 5 at exactly one point. What is the value of k?
1. Step-by-Step Formal Solution
- Equate: 2x^2 + 12x - (5 + k) = 0.
- Delta = 144 - 8(-5 - k) = 184 + 8k = 0.
- 8k = -184 => k = -23.
2. Fast Desmos / Calculator Shortcut
Graph both equations with slider k = -23. The line touches the parabola at exactly (-3, 7).
3. Distractor Trap Analysis
- Trap A (Sign Flip): Forgetting to reverse inequality or dropping negative sign.
- Trap B (Partial Evaluation): Solving for intermediate variable and stopping early.
- Trap C (Extreme Assumption): Over-inferring beyond textual evidence.
- Option Correct: Accurately satisfies all algebraic and logical constraints.
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