Question:For the quadratic equation 3x^2 - 12x + k = 0 with two distinct real roots, the sum of the...
GMAT Advanced Math : (Adv_Math) Questions
For the quadratic equation \(3\mathrm{x}^2 - 12\mathrm{x} + \mathrm{k} = 0\) with two distinct real roots, the sum of the two roots is four times the product of the two roots. What is the value of \(\mathrm{k}\)?
1. TRANSLATE the problem information
- Given information:
- Quadratic equation: \(3\mathrm{x}^2 - 12\mathrm{x} + \mathrm{k} = 0\)
- Two distinct real roots exist
- Sum of roots = 4 × Product of roots
- What this tells us: We need to express sum and product of roots in terms of k, then solve.
2. INFER the approach
- Key insight: Use Vieta's formulas instead of actually solving for the roots
- For any quadratic \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c} = 0\):
- Sum of roots = \(-\mathrm{b}/\mathrm{a}\)
- Product of roots = \(\mathrm{c}/\mathrm{a}\)
3. TRANSLATE using Vieta's formulas
For our equation \(3\mathrm{x}^2 - 12\mathrm{x} + \mathrm{k} = 0\) where a = 3, b = -12, c = k:
- Sum of roots = \(-(-12)/3 = 12/3 = 4\)
- Product of roots = \(\mathrm{k}/3\)
4. SIMPLIFY the given condition
Set up equation from "sum equals four times product":
- \(4 = 4 \times (\mathrm{k}/3)\)
- \(4 = 4\mathrm{k}/3\)
- Multiply both sides by 3: \(12 = 4\mathrm{k}\)
- Divide by 4: \(\mathrm{k} = 3\)
5. Verify our answer
Check that k = 3 gives two distinct real roots:
- Discriminant = \(\mathrm{b}^2 - 4\mathrm{ac} = (-12)^2 - 4(3)(3) = 144 - 36 = 108 \gt 0\) ✓
Answer: C) 3
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students try to actually solve the quadratic equation \(3\mathrm{x}^2 - 12\mathrm{x} + \mathrm{k} = 0\) for its roots instead of using the relationship between coefficients and roots. They get bogged down in the quadratic formula: \(\mathrm{x} = (12 \pm \sqrt{144 - 12\mathrm{k}})/6\), then struggle to express the sum and product relationships, often making algebraic errors in the complex expressions. This leads to confusion and guessing.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly set up \(4 = 4\mathrm{k}/3\) but make algebraic mistakes when solving. Common errors include forgetting to multiply both sides by 3 (getting \(\mathrm{k} = 1\)) or incorrectly dividing (getting \(\mathrm{k} = 12\)). These calculation errors may lead them to select Choice A (1) or guess among the remaining choices.
The Bottom Line:
This problem tests whether students recognize that Vieta's formulas provide an elegant shortcut. Students who try to "brute force" by finding actual roots miss the conceptual efficiency and often get lost in unnecessary algebra.