Question:\(\mathrm{y = 8x^2 + (4c - 2d)x - 20}\)In the given quadratic function, c and d are positive constants. The...
GMAT Advanced Math : (Adv_Math) Questions
\(\mathrm{y = 8x^2 + (4c - 2d)x - 20}\)
In the given quadratic function, \(\mathrm{c}\) and \(\mathrm{d}\) are positive constants. The graph of the equation in the xy-plane is a parabola with vertex \(\mathrm{(h, k)}\). If the x-coordinate of the vertex, \(\mathrm{h}\), can be written in the form \(\mathrm{m(d - 2c)}\), where \(\mathrm{m}\) is a constant, what is the value of \(\mathrm{m}\)?
1. TRANSLATE the problem information
- Given information:
- Quadratic function: \(\mathrm{y = 8x^2 + (4c - 2d)x - 20}\)
- c and d are positive constants
- Need to find m where \(\mathrm{h = m(d - 2c)}\)
- What this tells us: We need to find the x-coordinate of the vertex and express it in a specific form.
2. INFER the approach
- Since we need the x-coordinate of the vertex, we should use the vertex formula
- For any quadratic \(\mathrm{y = ax^2 + bx + c}\), the vertex x-coordinate is \(\mathrm{h = -b/(2a)}\)
- We'll need to manipulate our result algebraically to match the required form \(\mathrm{m(d - 2c)}\)
3. TRANSLATE the coefficients from standard form
- From \(\mathrm{y = 8x^2 + (4c - 2d)x - 20}\):
- \(\mathrm{a = 8}\)
- \(\mathrm{b = (4c - 2d)}\)
- \(\mathrm{constant = -20}\)
4. Apply the vertex formula
- \(\mathrm{h = -b/(2a)}\)
\(\mathrm{= -(4c - 2d)/(2 \times 8)}\)
\(\mathrm{= -(4c - 2d)/16}\)
5. SIMPLIFY the expression
- Distribute the negative sign: \(\mathrm{h = (-4c + 2d)/16}\)
- Factor out 2 from the numerator: \(\mathrm{h = 2(-2c + d)/16}\)
\(\mathrm{= 2(d - 2c)/16}\) - Reduce the fraction: \(\mathrm{h = (d - 2c)/8}\)
- Rewrite as: \(\mathrm{h = (1/8)(d - 2c)}\)
6. INFER the final answer
- Comparing \(\mathrm{h = (1/8)(d - 2c)}\) with the required form \(\mathrm{h = m(d - 2c)}\)
- We can see that \(\mathrm{m = 1/8}\)
Answer: 1/8 or 0.125
Why Students Usually Falter on This Problem
Most Common Error Path:
Missing conceptual knowledge: Not remembering the vertex formula \(\mathrm{h = -b/(2a)}\)
Students might try to complete the square or use other approaches, leading to unnecessary complexity and potential errors. Without the direct formula, they may get lost in algebraic manipulation or abandon the systematic approach entirely. This leads to confusion and guessing.
Second Most Common Error:
Weak SIMPLIFY skill: Making algebraic errors during the simplification from \(\mathrm{-(4c - 2d)/16}\) to \(\mathrm{(1/8)(d - 2c)}\)
Students might incorrectly distribute the negative sign, make errors when factoring, or struggle with fraction reduction. A common mistake is getting \(\mathrm{h = (2c - d)/8}\) instead of \(\mathrm{h = (d - 2c)/8}\), which would give \(\mathrm{m = -1/8}\). This may lead them to select an incorrect negative value if such options were provided.
The Bottom Line:
This problem tests both formula recall and algebraic manipulation skills. The key insight is recognizing that the vertex formula gives you the foundation, but you must then carefully manipulate the expression to match the required form. Success depends on methodical simplification without losing track of the target format.