SAT Math: If One Root of x² - 6x + 4 = 0 Is 3 + √5, What Is the Other Root?
By the conjugate root theorem, the second root is 3 - sqrt(5).
Exact Question Text
The equation x^2 - 6x + 4 = 0 has roots r1 and r2. If r1 = 3 + sqrt(5), what is the value of r2?
1. Step-by-Step Formal Solution
- Quadratic formula: x = (6 +- sqrt(36 - 16))/2 = (6 +- sqrt(20))/2 = 3 +- sqrt(5).
- If r1 = 3 + sqrt(5), then r2 = 3 - sqrt(5).
2. Fast Desmos / Calculator Shortcut
Type x^2 - 6x + 4 = 0 into Desmos to see the two intersection points: 3 + sqrt(5) and 3 - sqrt(5).
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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