The quadratic expression 4x^2 - 24x + 47 can be written in the form \(\mathrm{a}(\mathrm{x} - \mathrm{h})^2 + \mathrm{k}\) for...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
The quadratic expression \(4\mathrm{x}^2 - 24\mathrm{x} + 47\) can be written in the form \(\mathrm{a}(\mathrm{x} - \mathrm{h})^2 + \mathrm{k}\) for constants \(\mathrm{a}\), \(\mathrm{h}\), and \(\mathrm{k}\). What is the value of \(\mathrm{k}\)?
1. TRANSLATE the problem information
- Given: \(\mathrm{4x^2 - 24x + 47}\) needs to be written as \(\mathrm{a(x - h)^2 + k}\)
- Find: the value of \(\mathrm{k}\)
2. INFER the approach
- To find \(\mathrm{k}\), we need to complete the square
- Since the coefficient of \(\mathrm{x^2}\) is 4 (not 1), we should factor it out first to make completing the square easier
3. SIMPLIFY by factoring out the coefficient
Factor out 4 from the first two terms only:
\(\mathrm{4x^2 - 24x + 47 = 4(x^2 - 6x) + 47}\)
4. SIMPLIFY by completing the square
Inside the parentheses we have \(\mathrm{x^2 - 6x}\). To complete the square:
- Take half of the coefficient of x: \(\mathrm{-6 \div 2 = -3}\)
- Square this result: \(\mathrm{(-3)^2 = 9}\)
- Add and subtract 9 inside the parentheses:
\(\mathrm{4(x^2 - 6x + 9 - 9) + 47 = 4((x - 3)^2 - 9) + 47}\)
5. SIMPLIFY by distributing and combining
- Distribute the 4: \(\mathrm{4(x - 3)^2 - 4(9) + 47}\)
- Combine like terms: \(\mathrm{4(x - 3)^2 - 36 + 47 = 4(x - 3)^2 + 11}\)
Answer: A (11)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students often make calculation errors when completing the square, especially when finding \(\mathrm{(-6/2)^2 = 9}\). Some calculate this as 6 or 36 instead of 9.
If they use 36 instead of 9, they get:
\(\mathrm{4(x^2 - 6x + 36 - 36) + 47 = 4(x - 6)^2 - 144 + 47 = 4(x - 6)^2 - 97}\)
making \(\mathrm{k = -97}\), which isn't even an option. This leads to confusion and guessing.
Second Most Common Error:
Poor INFER reasoning: Students sometimes try to complete the square without factoring out the 4 first. This makes the algebra much more complicated and prone to errors.
Working directly with \(\mathrm{4x^2 - 24x + 47}\), they might try: \(\mathrm{4x^2 - 24x + ? + 47}\), leading to confusion about what value completes the square. This causes them to get stuck and randomly select an answer.
The Bottom Line:
The key insight is recognizing that factoring out the coefficient of \(\mathrm{x^2}\) first makes completing the square much cleaner and less error-prone. The arithmetic must be done carefully, especially when finding \(\mathrm{(-6/2)^2 = 9}\).