Value of a surd expression — Number System Aptitude Question
Value of a surd expression: If x = 1/(2 − √3), find the value of x³ − 2x² − 7x + 5. Answer and worked solution on CodeKairo.
- Topic: Number System
- Section: Quantitative Aptitude
- Difficulty: Hard
- Time target: 150 seconds
Question
If x = 1/(2 − √3), find the value of x³ − 2x² − 7x + 5.
Options
- 1
- 2
- 3
- 5
Show the answer and the worked solution
Correct answer: C. 3
Worked solution
- x = 1/(2 − √3) × (2 + √3)/(2 + √3) = (2 + √3)/(4 − 3) = 2 + √3.
- Then x − 2 = √3, so squaring gives x² − 4x + 4 = 3, i.e. x² − 4x + 1 = 0.
- Divide: x³ − 2x² − 7x + 5 = (x² − 4x + 1)(x + 2) + 3.
- The first term vanishes, leaving 3.
Answer: 3.
The idea
Rationalise, then convert the surd into a quadratic the variable satisfies. Reducing the higher polynomial by that quadratic avoids ever cubing a surd, which is where the arithmetic normally breaks down.
Shortcut. Estimate: x = 2 + √3 ≈ 3.732, so x² ≈ 13.93 and x³ ≈ 51.98. Then 51.98 − 27.86 − 26.12 + 5 ≈ 3.00, and every option is a whole number.
Common trap. 5 — assuming x² − 4x + 1 = 0 wipes out every x term and leaves the constant; (x² − 4x + 1)(x + 2) = x³ − 2x² − 7x + 2, which uses up 2 of the 5.