SAT Math: 3x² - 12x + c = 0 Has No Real Solutions, Find Least Integer Value of c

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

1. Standard form: a=3, b=-12, c=c.
2. Delta = b^2 - 4ac < 0 => 144 - 12c < 0 => c > 12.
3. 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.