SAT Math: 3x² - 12x + c = 0 Has No Real Solutions, Find Least Integer Value of c
Discriminant Delta = (-12)^2 - 4(3)(c) < 0 => 144 - 12c < 0 => c > 12. The least integer value is 13.
Exact Question Text
In the equation 3x^2 - 12x + c = 0, c is a constant. If the equation has no real solutions, what is the least possible integer value of c?
1. Step-by-Step Formal Solution
- Standard form: a=3, b=-12, c=c.
- Delta = b^2 - 4ac < 0 => 144 - 12c < 0 => c > 12.
- Smallest integer strictly greater than 12 is 13.
2. Fast Desmos / Calculator Shortcut
Type y = 3x^2 - 12x + c with slider c. Parabola lifts above x-axis when c > 12. At c = 13 there are 0 real roots.
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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