Question:Which of the following is a solution to the equation 9x^2 + 5x = 0?
GMAT Advanced Math : (Adv_Math) Questions
Question:
Which of the following is a solution to the equation \(9\mathrm{x}^2 + 5\mathrm{x} = 0\)?
1. INFER the solution strategy
- Given: \(\mathrm{9x^2 + 5x = 0}\)
- Key insight: Both terms contain x, so we can factor out x as a common factor
- This will be much simpler than using the quadratic formula
2. SIMPLIFY by factoring out the common factor
- Factor out x: \(\mathrm{9x^2 + 5x = x(9x + 5) = 0}\)
- Now we have a product that equals zero
3. INFER how to find the solutions
- Apply the zero product property: If \(\mathrm{x(9x + 5) = 0}\), then either \(\mathrm{x = 0}\) or \(\mathrm{9x + 5 = 0}\)
- This gives us two separate cases to solve
4. SIMPLIFY to find both solutions
- First solution: \(\mathrm{x = 0}\)
- Second solution: From \(\mathrm{9x + 5 = 0}\)
- Subtract 5: \(\mathrm{9x = -5}\)
- Divide by 9: \(\mathrm{x = -\frac{5}{9}}\)
5. APPLY CONSTRAINTS to select from answer choices
- Complete solution set: \(\mathrm{\{0, -\frac{5}{9}\}}\)
- Looking at the choices, only \(\mathrm{-\frac{5}{9}}\) appears as option (B)
Answer: B (\(\mathrm{-\frac{5}{9}}\))
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students correctly factor to get \(\mathrm{x(9x + 5) = 0}\) and know to solve \(\mathrm{9x + 5 = 0}\), but make an arithmetic error when dividing -5 by 9.
Instead of \(\mathrm{x = -\frac{5}{9}}\), they might flip the fraction and get \(\mathrm{x = -\frac{9}{5}}\), leading them to select Choice A (\(\mathrm{-\frac{9}{5}}\)).
Second Most Common Error:
Poor INFER reasoning: Students don't recognize the factoring approach and instead try to use the quadratic formula or attempt to solve by isolating x² first.
This leads to unnecessary complexity and potential calculation errors, causing confusion and possibly random guessing among the answer choices.
The Bottom Line:
The key insight is recognizing that when a quadratic has no constant term, factoring out the variable is always the simplest approach. Students who miss this strategy make the problem much harder than it needs to be.