If 4x = √5 + 2, simplify the expression x - 1/(16x) exactly in surd form and then select the correct value from the options given.

Introduction / Context:
This question involves algebraic manipulation with surds. You are given a simple linear relation between x and a surd expression, and then asked to simplify x - 1/(16x). The key is to express x in terms of the given surd and then compute the entire expression carefully, avoiding premature decimal approximations.

Given Data / Assumptions:

• 4x = √5 + 2.
• x is a real number.
• We must simplify x - 1/(16x) exactly, keeping surds in symbolic form.
• All operations are algebraic and use standard arithmetic rules.

Concept / Approach:
First solve 4x = √5 + 2 for x by dividing both sides by 4. Then compute 1/(16x) using that same expression for x. There is a useful observation that 16x = 4(4x), which simplifies the reciprocal. After expressing both x and 1/(16x) in terms of √5 and constants, subtract them and simplify using basic surd algebra.

Step-by-Step Solution:
From 4x = √5 + 2, divide both sides by 4 to get x = (√5 + 2)/4. Note that 16x = 4(4x) = 4(√5 + 2), so 1/(16x) = 1 / [4(√5 + 2)]. Thus x - 1/(16x) = (√5 + 2)/4 - 1/[4(√5 + 2)]. Factor out 1/4: (1/4)[(√5 + 2) - 1/(√5 + 2)]. Compute (√5 + 2) - 1/(√5 + 2) by writing over a common denominator: [(√5 + 2)^2 - 1] / (√5 + 2). Expand (√5 + 2)^2 = 5 + 4√5 + 4 = 9 + 4√5. Then (9 + 4√5 - 1) = 8 + 4√5. So (√5 + 2) - 1/(√5 + 2) = (8 + 4√5) / (√5 + 2). Factor 4: 4(2 + √5)/(√5 + 2) = 4 because numerator and denominator are equal. Finally x - 1/(16x) = (1/4) * 4 = 1.
Verification / Alternative check:
You may also observe directly that if 4x = √5 + 2, then 1/(4x) = 1/(√5 + 2). The expression x - 1/(16x) can be rewritten in terms of 4x and its reciprocal, leading to the same simplification. Numerical verification using approximate values of √5 confirms that both sides are very close to 1.

Why Other Options Are Wrong:
Option a ( 2√5 ) and option c ( 4 ) arise from incomplete simplification of the surd or missing the factor of 1/4. Option b ( -1 ) is the negative of the true result and suggests a sign error during subtraction. Option e ( 0 ) would only be obtained if x and 1/(16x) were equal, which they are not for this value of x.

Common Pitfalls:
Learners sometimes attempt to rationalise denominators too early or mix up 4x and 16x, leading to wrong reciprocals. It is also easy to misexpand (√5 + 2)^2 if you forget the middle term 2·√5·2 = 4√5. Writing each algebraic step clearly helps avoid these mistakes.

Final Answer:
The simplified exact value of x - 1/(16x) is 1.