If 3x^2 - 18x - 15 = 0, what is the value of x^2 - 6x?
GMAT Advanced Math : (Adv_Math) Questions
If \(3\mathrm{x}^2 - 18\mathrm{x} - 15 = 0\), what is the value of \(\mathrm{x}^2 - 6\mathrm{x}\)?
1. TRANSLATE the problem information
- Given equation: \(\mathrm{3x^2 - 18x - 15 = 0}\)
- Target: Find the value of \(\mathrm{x^2 - 6x}\)
2. INFER the most efficient approach
- Key insight: We don't need to solve for x completely
- Notice that if we divide the given equation by 3, we'll get terms that look like our target expression
- Strategy: Manipulate the equation to isolate \(\mathrm{x^2 - 6x}\) directly
3. SIMPLIFY by dividing the entire equation by 3
- \(\mathrm{3x^2 - 18x - 15 = 0}\)
- Divide every term by 3: \(\mathrm{x^2 - 6x - 5 = 0}\)
- Now we have the exact expression we're looking for: \(\mathrm{x^2 - 6x}\)
4. SIMPLIFY to isolate the target expression
- From \(\mathrm{x^2 - 6x - 5 = 0}\)
- Add 5 to both sides: \(\mathrm{x^2 - 6x = 5}\)
Answer: 5
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students immediately try to solve the quadratic equation completely for x values instead of recognizing they can work directly with the expression they need.
They might use the quadratic formula: \(\mathrm{x = \frac{18 \pm \sqrt{324 + 180}}{6} = \frac{18 \pm \sqrt{504}}{6}}\), then try to substitute these complex values back into \(\mathrm{x^2 - 6x}\). This creates unnecessary computational complexity and often leads to arithmetic errors or giving up due to the messy calculations.
This leads to confusion and guessing.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly identify that they should divide by 3, but make sign errors or forget to divide all terms consistently.
For example, they might get \(\mathrm{x^2 - 6x + 5 = 0}\) instead of \(\mathrm{x^2 - 6x - 5 = 0}\), leading them to conclude \(\mathrm{x^2 - 6x = -5}\) instead of 5.
This may lead them to select an incorrect negative answer if available in the choices.
The Bottom Line:
This problem rewards strategic thinking over computational power. The key insight is recognizing that sometimes the most direct path to an answer involves manipulating the given equation rather than solving it completely.