If x = 2 + square root of 3, evaluate the value of the expression (x^2 - x + 1) divided by (x^2 + x + 1).

Introduction / Context:

This aptitude question checks your understanding of algebraic simplification with surds and rational expressions. You are given a specific value of x that involves a square root term and asked to evaluate a rational expression built from x^2 - x + 1 and x^2 + x + 1. Such questions are common in competitive exams because they test your ability to manipulate expressions without getting lost in long decimal calculations.

Given Data / Assumptions:

• x = 2 + square root of 3.
• We need to find the value of (x^2 - x + 1) / (x^2 + x + 1).
• x is a real number and square root of 3 is the positive root.

Concept / Approach:

The key idea is to compute x^2, then form the required numerator and denominator. Because the expression is symmetric in x, the surd terms will often cancel, giving a rational answer. Working symbolically is better than converting square root of 3 into a decimal.

Step-by-Step Solution:

Verification / Alternative check:

You can verify the result by approximating sqrt(3) as 1.732. Then x is about 3.732. Evaluating the expression numerically gives a value close to 0.6, which matches 3/5.

Why Other Options Are Wrong:

The value 2/3 is larger than 3/5 and would arise from incorrect cancellation. The fraction 3/4 comes from mixing up coefficients when simplifying. The option 4/5 comes from cancelling only part of the surd term. The option 1/2 corresponds to a rough estimate that ignores exact algebraic structure. None of these match the exact simplification.

Common Pitfalls:

Common mistakes include squaring 2 + sqrt(3) incorrectly, forgetting the middle term 2*2*sqrt(3), or trying to rationalize by multiplying numerator and denominator by a wrong conjugate. Another frequent error is cancelling (2 + sqrt(3)) directly without first factoring it out properly.

Final Answer:

The exact value of the expression (x^2 - x + 1) / (x^2 + x + 1) is 3/5.