x^2 - bx + 15 = 0 In the equation above, b is a positive constant. If the equation has...
GMAT Advanced Math : (Adv_Math) Questions
\(\mathrm{x^2 - bx + 15 = 0}\)
In the equation above, \(\mathrm{b}\) is a positive constant. If the equation has two positive integer solutions and the difference between the larger solution and the smaller solution is \(\mathrm{2}\), what is the value of \(\mathrm{b}\)?
1. TRANSLATE the problem information
- Given information:
- Quadratic equation: \(\mathrm{x^2 - bx + 15 = 0}\)
- b is positive
- Two positive integer solutions
- Difference between larger and smaller solution is 2
- Find the value of b
2. INFER the strategic approach
- Since we need relationships between the roots and coefficient b, Vieta's formulas are perfect here
- Let the two solutions be r and s where \(\mathrm{r \gt s}\)
- We can use both the sum and product relationships
3. Apply Vieta's formulas
For \(\mathrm{x^2 - bx + 15 = 0}\):
- Sum of roots: \(\mathrm{r + s = b}\)
- Product of roots: \(\mathrm{rs = 15}\)
4. CONSIDER ALL CASES for factor pairs
Since both roots are positive integers with product 15, find all factor pairs:
- \(\mathrm{1 × 15 = 15}\)
- \(\mathrm{3 × 5 = 15}\)
5. APPLY CONSTRAINTS using the difference condition
Check which pair has difference 2:
- \(\mathrm{15 - 1 = 14 ≠ 2}\) ✗
- \(\mathrm{5 - 3 = 2}\) ✓
Therefore: \(\mathrm{r = 5, s = 3}\)
6. Calculate b
\(\mathrm{b = r + s = 5 + 3 = 8}\)
Answer: 8
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't recognize that Vieta's formulas provide the direct connection between roots and coefficients. Instead, they might try to guess and check values of b or attempt to factor without a systematic approach.
This leads to confusion and abandoning systematic solution, resulting in guessing.
Second Most Common Error:
Poor APPLY CONSTRAINTS execution: Students find the factor pairs of 15 but forget to check the difference condition. They might immediately conclude that \(\mathrm{b = 1 + 15 = 16}\) using the first factor pair they find.
This may lead them to select an incorrect answer if 16 were among the choices.
The Bottom Line:
This problem requires connecting abstract algebraic relationships (Vieta's formulas) with concrete number properties (factor pairs and differences). Students who miss the Vieta's formulas connection often get lost trying random approaches.