Question:The quadratic expression 3x^2 - 5x + 2 is rewritten in the form \((3\mathrm{x} + \mathrm{k})(\mathrm{x} - 1)\), where k...
GMAT Advanced Math : (Adv_Math) Questions
The quadratic expression \(3\mathrm{x}^2 - 5\mathrm{x} + 2\) is rewritten in the form \((3\mathrm{x} + \mathrm{k})(\mathrm{x} - 1)\), where \(\mathrm{k}\) is a constant.
What is the value of \(\mathrm{k}\)?
Enter your answer as an integer.
No answer choices - enter your answer as an integer (grid-in).
1. TRANSLATE the problem information
- Given: \(\mathrm{3x^2 - 5x + 2 = (3x + k)(x - 1)}\)
- Find: The value of k
2. INFER the solution approach
- Since these expressions are equal, I can expand the right side and compare coefficients
- Alternatively, I could factor the left side directly and match forms
3. SIMPLIFY by expanding the factored form
- Expand \(\mathrm{(3x + k)(x - 1)}\):
- First: \(\mathrm{3x × x = 3x^2}\)
- Outer: \(\mathrm{3x × (-1) = -3x}\)
- Inner: \(\mathrm{k × x = kx}\)
- Last: \(\mathrm{k × (-1) = -k}\)
- Combine: \(\mathrm{3x^2 - 3x + kx - k = 3x^2 + (k - 3)x - k}\)
4. SIMPLIFY by comparing coefficients
- Set expanded form equal to original:
\(\mathrm{3x^2 + (k - 3)x - k = 3x^2 - 5x + 2}\)
- Compare x-coefficients: \(\mathrm{k - 3 = -5}\)
Solve: \(\mathrm{k = -5 + 3 = -2}\)
- Verify with constant terms: \(\mathrm{-k = 2}\)
This gives \(\mathrm{k = -2}\) ✓
Answer: -2
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Sign errors during expansion, particularly with the constant term.
Students often expand \(\mathrm{(3x + k)(x - 1)}\) as \(\mathrm{3x^2 - 3x + kx + k}\) instead of \(\mathrm{3x^2 - 3x + kx - k}\), forgetting that \(\mathrm{k × (-1) = -k}\). When they compare constant terms, they get \(\mathrm{+k = 2}\) instead of \(\mathrm{-k = 2}\), leading to \(\mathrm{k = 2}\) instead of \(\mathrm{k = -2}\).
Second Most Common Error:
Inadequate SIMPLIFY execution: Arithmetic mistakes when solving \(\mathrm{k - 3 = -5}\).
Students correctly set up the coefficient equation but make calculation errors, getting \(\mathrm{k = -5 - 3 = -8}\) or \(\mathrm{k = -5/3}\), leading to confusion about which form to enter as their answer.
The Bottom Line:
This problem requires careful attention to signs during algebraic expansion and systematic coefficient comparison. The multiple verification paths (x-coefficient and constant term) actually help catch errors if students use both methods.